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THE VERTICAL PROPAGATION OF INERTIAL WAVES
IN THE OCEAN
by
KEVIN DOUGLAS LEAMAN
B.S., University of Michigan(1970)
M.S., University of Michigan(1972)
SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
and the
WOODS HOLE OCEANOGRAPHIC INSTITUTION
June, 1975
Signature of Author........7.- ....... ......Joint Program in Oceanography, MassachusettsInstitute of Technology - Woods HoleOceanographic Institution, and Department ofEarth and Planetary Sciences, and Departmentof Meteorology, Massachusetts Institute ofTechnology, June 1975
I I
Certified by.... .S .v r;..v; .- r. r ~ . " . . iV w.. *sor
SKThesis Supervisor
Accepted by.. ..-.. ... . .. ... ., .. . .. ...ChairFmi , Joint Oceanogra y Committee in theEarth Sciences, Massachusetts Institute ofTechnology - Woods Hole Oceanographic.Institution
2
THE VERTICAL PROPAGATION OF INERTIAL WAVES IN THE OCEAN
by
Kevin Douglas Leaman
Submitted to the Massachusetts Institute of Technology-Woods HoleOceanographic Institution Joint Program in Oceanography in June,1975, in partial fulfillment of the requirements for the degreeof Doctor of Philosophy.
ABSTRACT
A set of vertical profiles of horizontal ocean currents, ob-tained by electro-magnetic profilers in the Atlantic Ocean southwestof Bermuda in the spring of 1973, has been analyzed in order to studythe vertical structure and temporal behavior of internal waves, par-ticularly those with periods near the local inertial period. An im-portant feature of the observed structure is the polarization of hor-izontal velocity components in the vertical. This polarization, alongwith temporal changes of the vertical wave structure seen in a timeseries of profiles made at one location, has been related to the di-rection of vertical energy flux due to the observed waves. Whereasthe observed vertical phase propagation can be affected by horizontaladvection of waves past the point of observation, the use of wave po-larization to infer the direction of vertical energy propagation hasthe advantage that it is not influenced by horizontal advection. Theresult shows that at a location where profiles were obtained oversmooth topography, the net energy flux was downward, indicating thatthe energy sources for these waves were located at or near the seasurface. An estimate of the net, downward energy flux (- .2 - .3erg/cm2/sec) has been obtained. Calculations have been made whichshow that a frictional bottom boundary layer can be an important en-ergy sink for near-inertial waves. A rough estimate suggests thatthe observed, net, downward energy flux could be accounted for byenergy losses in this frictional boundary layer. A reflectioncoefficient for the observed waves as they reflect off the bottomhas been estimated.
In contrast, some profiles made over a region of rough topog-raphy indicate that the rough bottom may also be acting to generatenear-inertial waves which propagate energy upward.
Calculations of vertical flux of horizontal kinetic energy,using an empirical form for the energy spectrum of internal waves,show that this vertical flux reaches a maximum for frequencies 10% -20% greater than the local inertial frequency. Comparison with pro-filer velocity data and frequency spectra supports the conclusionthat the dominant waves had frequencies 10% - 20% greater than theinertial frequency. The fact that the waves were propagating energyin the vertical is proposed as the reason for the observed frequencyshift.
Finally, energy spectra in vertical wave number have beencalculated from the profiles in order to compare the data with anempirical model of the energy density spectrum for internal wavesproposed by C. Garrett and W. Munk (1975). The result shows thatalthough the general shape and magnitude of the observed spectrumcompares well with the empirical model, the two-sided spectrum isnot symmetric in vertical wave number. This asymmetry has beenused to infer that more energy was propagating downward than upward.These calculations have also been used to obtain the coherence be-tween profiles made at the same location, but separated in time(the so-called dropped, lagged, rotary coherence). This coherenceis compared with the aforementioned empirical model. The coherenceresults show that the contribution of the semidiurnal tide to theenergy of the profiles is restricted to long vertical wave lengths.
Thesis Supervisor: Dr. Thomas B. SanfordTitle: Associate Scientist
ACKNOWLEDGEMENTS
I wish to thank above all Dr. Thomas B. Sanford of the
Woods Hole Oceanographic Institution for providing the data on
which this report is based, and for many helpful comments and
suggestions. Initial acquisition and processing of data obtained
by the electro-magnetic profiler was successful primarily due to
the efforts of R. G. Drever, E. Denton, A. Bartlett, J. Dunlap
and J. Stratton, all of the above institution. Support for the
experiment which is described in this report was provided by the
Office of Naval Research under contracts N00014-66-C-0241, NR 083-
004 and N000 14-74-C-0262, NR 083-004.
For my personal support as a graduate student I thank the
National Science Foundation for its award to me of a graduate
fellowship for three years. I also thank the Office of Naval
Research (under the above contracts) for providing part of my
support.
TABLE OF CONTENTS
Page
Abstract
Acknowledgements
List of Figures
Chapter I Introduction and Historical Review
Chapter II Description and Analysis of Data from theFive-day Time Series of Profiles
a.) Dominance of inertial-period motions inEMVP profiles--the "mirror-imaging" of.profiles
b.) Low-frequency "geostrophic" profiles
c.) High-frequency velocity components inthe upper 2.5 km as a function of time
d.) The average perturbation kinetic energyprofile, and a smoothed profile of Brunt-V*is*l* frequency
e.) The stretching of the vertical coordinateand the normalization of velocity by themean Brunt-Visal profile
f.) Spectral decomposition of stretched pro-files
g.) Paired profiles
Chapter III Vertical Polarization and the Vertical Propa-gation of Internal-inertial Waves
a.) Summary of data
b.) The vertical spatial behavior of hori-zontal internal wave velocity components
10
21
21
23
27
32
36
42
54
57
57
58
Chapter III
Chapter IV
Chapter V
Chapter VI
Chapter VII
Appendix A
Appendix B
Appendix C
Appendix D
Bibliography
c.) Polarization in the deep water
d.) Comparison with meteorological observa-tions
The Use of the Ratio of Clockwise to Counter-clockwise Energy as a Reflection Coefficient.Reflection of Internal Waves from a SmoothBottom
a.) Calculation of a reflection coefficientfor the observed waves
.b.) A model for the reflection of internalwaves from a smooth bottom
Comparison of Vertical Wave Number Spectrawith Theoretical Models. Computation ofDropped, Time-lagged, Rotary Coherence (DLRC).The Vertical Energy Flux of Internal-inertialWaves
a.) Vertical wave number spectra
b.) Dropped, time-lagged, rotary coherenceof profiles
c.) The vertical energy flux of near-inertialwaves
Some Observations over Rough Topography
Conclusions, a Further Discussion of Some ofthe Data and Recommendations for FutureExperiments
Profile Locations and Times
Use of Electro-magnetic Velocity Profilersto Measure Internal Waves
Accuracy and Precision of Observations
A More General Treatment of the Partition ofEnergy Between Clockwise and CounterclockwiseSpectra
Biographical Note
Page71
89
98
107
119
126
142
146
159
163
170
174
LIST OF FIGURES
FigureNumber Page
1. Electro-magnetic velocity profiler 16
2. Bathymetry of the MODE region. Location of central 18mooring time series is shown by an X. The ridge overwhich rough topography profiles were made is circled.
3. Paired drops, 219D and 221D, made at the same loca- 22tion and with a time separation of one half of aninertial period.
4. Average profile of the east velocity component during 25the five-day time series.
5. Average profile of the north velocity component during 26the five-day time series.
6. Contours of the east velocity component during the 28five-day time series. Negative east components areindicated by hatched regions.
7. Contours of the north velocity component during the 29five-day time series. Negative north components areindicated by hatched regions.
8. Average profiles of high-frequency horizontal kinetic 33energy and Brunt-VaislIN frequency.
9. Stretched and unstretched vertical pressure coordinates. 39
10. East component of profile 219D as originally obtained 40(a), after removal of the mean profile (b), and afterstretching and normalizing(c).
11. North component of profile 219D as originally obtained 41(a), after removal of the mean profile (b), and afterstretching and normalizing (c).
12. Total spectrum (east and north energy) for 9 down pro- 43files.
FigureNumber Page
13. Clockwise (CW) and counterclockwise (ACW) spectra for 459 down profiles. In the text, CW is C(m') and ACW isA(m').
14. Spectra of east and north velocity components for 9 48down profiles.
15. Total spectrum (east and north energy) for 9 up pro- 49files.
16. Clockwise (CW) and counterclockwise (ACW) spectra 50for 9 up profiles.
17. Spectra of east and north velocity components for 9 51up profiles.
18. Simultaneous paired drops, 235U and 236U, separated 55horizontally by 4.8 km.
19. Horizontal coordinate rotation (see equations 111-2). 59
20. Direction of rotation of the horizontal velocity vec- 59tor of an internal wave for positive and negativevertical wave numbers. The curved arrow shows thedirection in which V rotates with increasing z.
21. Propagation diagram for internal waves. Cg is the 62group velocity vector, k is the wave number vector,and V is a vector representing water velocity.
22. Reflection coefficient A(m')/C(m'). 79
23. Comparison of theoretical and observed total energy 96spectra in stretched vertical wave number.
24. Dropped, lagged, rotary cospectrum (normalized) be- 101tween velocity components U1 and Us (see text for afurther description of Figures 24 to 27).
25. Dropped, lagged, rotary cospectrum (normalized) be- 102tween velocity components VI and V3 .
26. Dropped, lagged, rotary quadrature spectrum (normal- 103ized) between velocity components U, and U,.
FigureNumber Page
27. Dropped, lagged, rotary quadrature spectrum (normal- 104ized) between velocity components V1 and V3.
28. Frequency dependence of the vertical flux of hori- 114zontal kinetic energy, using a theoretical form forthe energy spectrum of the internal wave field.
29. Frequency spectrum, obtained by averaging 12 spectra 117computed from current meter records at 1500 m inthe MODE region. Energy density units are(cm/sec)2/cph.
30. Smoothed hodograph of profile 190D between 3000 dbar 122and 4500 dbar. Depths are in hundreds of decibars.
31. Smoothed hodograph of profile 192D between 3000 dbar 123and 4500 dbar. Depths are in hundreds of decibars.
Chapter I Introduction and Historical Review
For many years the subject of internal waves has been of in-
tense interest to meteorologists, aeronomists, and oceanographers.
These internal waves are oscillations in the atmosphere or ocean
that are supported by buoyancy and pressure forces, and also by the
fictitious Coriolis force introduced by the rotation of the earth.
Oscillations having time and length scales appropriate to internal
waves have been observed at many locations in the ocean. In the
atmosphere, motions having time and space scales appropriate to
internal waves have also been observed. However, until recently,
meteorologists and oceanographers have been studying internal wave
phenomena from decidedly different points of view. This has been
brought about primarily by the different types of instrumentation
used in the two disciplines. Observations of atmospheric internal
waves are discussed first.
In the years since World War II, several methods have been
developed by meteorologists for observing variables such as wind
and temperature in the atmosphere. Wind measurements, which are
of primary interest in the present study, have been performed using
the following methods: tracking the distortion of smoke (actually
sodium) trails left by high altitude rockets; tracking of radar re-
flecting balloons with ground-based radar; visual tracking of meteor
trails by telescopes; tracking chaff (strips of aluminized reflect-
ing material) with ground-based radar; and radio-echo tracking of
meteor trails. All of these methods have their disadvantages.
Rockets, for example, are expensive to use, and they have not
usually been applied to rapidly repeated measurements in the at-
mosphere. Visual and radio-echo tracking of meteors depend, of
course, on a sufficient number of meteors entering the upper at-
mosphere. Except in rare cases when meteor showers occur, the
number of meteors in the atmosphere has usually been insufficient
to study high-frequency oscillations in the wind field, such as
internal waves. More recently, tracking of ascending, radar re-
flecting balloons has provided better measurements of atmospheric
winds.
All of these methods, although diverse in concept, have in
common the fact that they provide profiles of horizontal atmos-
pheric winds as a function of altitude. Many examples of these
profiles have been presented in the literature (see, for example,
the paper by L. Broglio in "Proceedings of the First International
Symposium on Rocket and Satellite Meteorology," in which results
from rocket profiles are presented, and the radar balloon observa-
tions of Endlich, Singleton and Kaufman (1969)). Thus, the atten-
tion of meteorologists and aeronomists has historically been di-
rected toward observing the vertical structure of the atmospheric
wind field. Attempts at using repeated profiles to determine the
amount of energy contributed by high-frequency (internal wave and
turbulent) oscillations at different time scales or frequencies
have been notably less successful than the measurements of the
profiles themselves. These attempts have usually consisted of try-
ing to fit diurnal and semidiurnal period sinusoids to the winds
observed at a given altitude (Elford and Robertson, 1953; Smith,
1960).
The situation in the oceans has been substantially different.
Until recently, the primary tools which oceanographers have used to
observe oscillations with frequencies in the internal wave range
have been fixed-location current meters, temperature sensors, and
neutrally buoyant floats. The current meter allows one to measure,
at a fixed point in the ocean, the horizontal current vector as a
function of time. If current meters are placed at many locations
in a part of the ocean (by being put on moorings), then time meas-
urements in a spatial array are obtained. These arrays allow one
to obtain good temporal resolution of velocity, but only at a
limited number of points in space.
Current meters have provided a large volume of data on the
temporal behavior of the ocean. Fofonoff (1969) showed that the
behavior of horizontal velocity components and temperature in the
internal wave frequency band (those frequencies corresponding to
periods between one half of the local pendulum day, 1/(2sin(Lat.))
days, and the local stable oscillation period of a particle of
water, or the Brunt-VIisali period) could be reasonably well re-
lated to the expected behavior of internal waves in a linear model.
Webster (1968) summarizes many of the current meter observations
that have been used to study internal waves and other high-frequency
oceanic phenomena. Most of these current meter records have shown
that the dominant contributions to horizontal kinetic energy spec-
tra come from frequencies near the local inertial frequency, cor-
responding to the aforementioned period of one half of a pendulum
day, and from the tides.
Temperature observations have been made using several dif-
ferent methods. Hauritz, Stommel and Munk (1959) observed temp-
erature oscillations as a function of time at two fixed depths,
50 and 550 meters, near Bermuda. These data yielded frequency
spectra for the temperature oscillations.
Another method, employed by Charnock (1965) and LaFond and
LaFond (1971), is to tow a weighted thermistor chain behind a
vessel. If the speed of the vessel is large compared to the hori-
zontal phase propagation speed of the waves being observed, spec-
tral analysis of the temperature trace obtained at a gi.ven depth
gives a horizontal wave number spectrum of the observed isotherm
displacements. Similar observations can be obtained by towing an
isotherm-following "fish" behind a research vessel (Katz, 1973,
1975).
The use of neutrally buoyant floats has also provided data
on internal waves. Pochapsky (1972) used such floats to observe
oscillations of temperature and horizontal velocity with periods
in the internal wave range. Voorhis (1968) obtained measurements
of vertical motion with a neutrally buoyant float which rotates in
the presence of vertical flow past the float. The rate of rota-
tion of the float gives an estimate of this vertical flow.
Many of the observations described above (as well as others
which have not been mentioned) have been gathered together by
Garrett and Munk (1972, 1975) in an attempt to model the energy
spectrum of internal waves in frequency and horizontal wave number.
The model depends not only on individual spectra of temperature or
velocity, but also on the correlation, or coherence, between obser-
vations made at different points of a spatial array. Coherence
observations have also been gathered together and described by
those authors (Garrett and Munk, 1972, 1975).
A major difficulty with this type of model construction has
been lack of knowledge concerning the vertical structure of the
wave field. This vertical structure is difficult to determine
using velocity (or temperature) measurements from moorings, since
only a limited number of instruments can be placed on any one
mooring. Thus, vertical profiles of horizontal velocity, which
have been the most common method of observation in the atmosphere,
have been the least common type of observation in oceanographic
work.
Recently, however, several methods have been developed that
allow the oceanographer to obtain vertical profiles of horizontal
current in the ocean that are analogous to profiles obtained in the
atmosphere. One such method involves the use of acoustic trans-
ponders located on the ocean bottom to track a float as it descends
through the water column (Rossby, 1969; Pochapsky and Malone, 1972).
Another method uses a moored vertical cable with inclinom-
eters located at various points along the cable (McNary, 1968).
Horizontal currents cause the cable to distort. This distortion,
as measured by the inclinometers, can then be converted to values
of the horizontal velocity component along the cable. A third
method uses horizontal electrical currents induced by the flow of
ocean water through the terrestrial magnetic field (Sanford, 1971).
The device which measures these electrical currents is called an
electro-magnetic velocity profiler, or EMVP. A picture of this
device, developed by T. Sanford and R. Drever of the Woods Hole
Oceanographic Institution, is shown in Figure 1. The EMVP is re-
leased from a research vessel, and as it falls to the ocean bottom
it continuously records the horizontal component of electrical
current, as well as temperature, conductivity, and pressure. When
the EMVP reaches the bottom, it reverses and ascends through the
water column, again recording the above variables. There are no
hard connections (cables, etc.) between the research vessel and
the EMVP: the only communication from EMVP to ship is through
acoustic telemetry. All data are recorded on 7-track magnetic
tape within the EMVP. Upon return to the ship, the data on the
tape is reformatted onto 9-track tape, and several computer programs
are used to convert the internally recorded variables to their
physical counterparts, such as temperature. Another routine, using
a conversion equation, converts horizontal electrical current den-
sity to horizontal water flow relative to an unknown, but depth
independent, horizontal velocity. A further description of the
EMVP and its operation can be found in Sanford, et. al. (1974).
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The horizontal velocity component is not the only quantity
that has been measured by vertical profiling. Vertical profiles
of temperature, conductivity and salinity have been obtained by
raising and lowering a CTD (conductivity-temperature-depth sensor).
Such data have been presented by Hayes (1975) and Hayes, et. al.
(1975).
We turn now to a description of the experiment which is
discussed in this report. In the spring of 1973 a large-scale
oceanographic experiment called MODE (Mid-Ocean Dynamics Experi-
ment) was carried out in the Atlantic Ocean, southwest of Bermuda.
A map showing the bathymetry in the region of this experiment is
shown in Figure 2. The EMVP was used to take a large number of
vertical profiles during the experiment. The locations, dates and
times of all profiles are given in Appendix A. We are going to
concentrate particularly on two subsets of all the profiles. The
first subset consists of a time series of repeated profiles made
at the center of the experiment (69* 40' W, 27* 59' N). This will
be called the "central mooring time series," since the central
mooring of the MODE experiment was located near these coordinates.
This time series, made up of 20 profiles, lasted from June 11-15,
1973. (These profiles are designated by an asterisk in Appendix A).
As can be seen from Figure 2, the location of the series is over a
region of smooth bottom topography. The second subset is made up
of some profiles taken around a ridge east of the time series, in a
region of rough topography. These locations are shown in Figure 2.
Al i I ~I .~I I . I a I30
AMES indicate seamounts MODE Region Bathymetry byand minimounts
) .f ,, P, Bush)epths in corrected meters o: (
AOML/NOAA/ _ -29
4I /' 'VANGUARDS;VOLKSTWAMANN .
O MODE 44602
-28
G UL .SWALLOW "-...L",
26
Figure 2. Bathymetry of the MODE region. Location of centralmooring time series is shown by an X. The ridge overwhich rough topography profiles were made is circled.
Any profile that has ever been made by the EMVP has shown
two dominant contributions to the velocity structure in the verti-
cal. The first contribution is a low-frequency part, which does
not change appreciably from one profile to the next at the same
location, and which corresponds well with current shear derived by
geostrophic shear calculations in that region. The second part is
a high-frequency contribution which is evidently due to internal
waves, particularly those with frequencies near the local inertial
frequency. It is this high-frequency part that will be of main
interest in what follows.
Chapter II gives a description and analysis of the velocity
data obtained fr',m the five-day time series of profiles. Plots of
mean flow and average perturbation horizontal kinetic energy versus
depth are given, as are contour plots showing the evolution of
features in the velocity field in time, and spectra of 'the high-
frequency profile structure in vertical wave number.
One of the most interesting results obtained from the spec-
tral analysis in Chapter II is that there is a definite elliptical
polarization of the high-frequency waves in the profiles. That is,
the horizontal current vector in a profile (after the mean has been
removed) clearly tends to rotate with depth. This polarization and
its relation to the vertical propagation of internal wave energy is
described in Chapter III. In Chapter III, comparisons will be made
between our observations and similar observations made in the atmos-
phere.
Chapter IV describes the reflection of internal waves from
a smooth bottom. The spectra presented in Chapter II are used to
attempt to calculate reflection coefficients for these waves.
Chapter V presents a comparison of the vertical wave number
spectra in Chapter II to the theoretical models of Garrett and
Munk (1972, 1975). The coherence of velocity components between
pairs of profiles separated by a time lag is also presented. Ver-
tical energy flux of the observed waves is discussed in more detail.
Chapter VI discusses the profiles obtained over rough topog-
raphy. Chapter VII provides some conclusions, a further discussion
of some of the data, and recommendations for future experiments.
~Y_~XIL________IY_____ _~1_1 I___ ~I .~
Chapter II Description and Analysis of .Data from the Five-day
Time Series of Profiles
It was mentioned in the introduction that EMVP data is his-
torically more akin to meteorological measurements in the upper at-
mosphere than to oceanographic data. In particular, profiler data
emphasizes the vertical structure of the ocean or atmosphere, while
until recently most oceanographic work has emphasized the temporal
behavior of oceanic motions. This chapter will present, in various
forms, horizontal velocity data obtained during the five-day series
of profiles, and will describe some of the methods used to reduce
and analyze this data. Succeeding chapters are devoted to the
interpretation of results obtained from this analysis. The reader
may refer to Appendix B for a brief discussion of internal wave
measurements made by the EMVP and to Appendix C for a discussion of
the accuracy and precision of data presented in the following sec-
tions.
a.) Dominance of inertial-period motions in EMVP profiles--the"mirror-imaging" of profiles
Figure 3 shows the east and north components of two profiles,
219D and 221D. (The letter D signifies that the down portion of a
drop was used. The letter U will be used to denote the up portion
of a drop.) The time separation between the two profiles is approx-
imately one half of an inertial period. The vertical sampling rate
is one point every 10 dbar. It is evident from the figure that,
~ 11~1111~- 1_1_L~L1_-i (--III--L~-._L~1-~i-~__l;_l_ ___-QI~ ilY
219D & 221D
EAST5 (CM/SI) NORTH, (CM/S)
Paired drops, 219D and 221D, made at the same locationand with a time separation of one half of an inertialperiod.
Cr)U4l
Figure 3.
VELOCITY PROFILES
superimposed on a low-frequency shear in the profiles, there is a
considerable amount of high-frequency activity. This high-frequen-
cy activity is evident in the almost total reversal of the velocity
structure over the time interval. If we remove the low-frequency
shear of the profiles (by averaging all drops in the time series
and subtracting the low-frequency shear profile from each drop sep-
arately), we find that the high-frequency part that is left over
is negatively correlated in profiles made one half of an inertial
period apart. This negative correlation, or "mirror-imaging," is
an indication that the EMVP profiles are strongly dominated in the
high frequencies by motions with periods close to the local iner-
tial period. It will be shown later that some of the observed
variability is also contributed by the semidiurnal tide.
b.) Low-frequency "geostrophic" profiles
For ease of discussion we will call oscillations with fre-
quencies less than the local inertial frequency "low-frequency"
oscillations, or the "geostrophic" part of the flow. Similarly,
oscillations with frequencies greater than or equal to the inertial
frequency will be called "high-frequency," or "internal wave"
oscillations. It should be clear that these terms are used solely
to simplify the description of the data. We do not intend to
imply that all motions with frequencies less than inertial are in
geostrophic balance. Also, high-frequency motions are not necessar-
ily due to internal waves. A certain amount of high-frequency en-
ergy may be contributed by turbulence, for example. It is known,
H~^-~" ~L I^~" ~'-Y~--eXlf~L~ ~-LII IIC-.-~C-~CY---
however, that there is a "well,." or depression in frequency spectra
obtained in the MODE region (see Figure 29). This well exists in
the range of periods from the inertial period to periods of several
days. Therefore, averaging velocity data over several days will
partition the profile data into high-frequency and low-frequency
parts. In order to characterize more fully this partition of en-
ergy between the low-frequency "geostrophic" and high-frequency
"internal wave" parts of the profiles, we have averaged all 20 pro-
files in the time series. (The 20 profile time series occupies a
time interval of about 103 hours.) This averaging was done simply
by removing the "barotropic," or depth-averaged, part of each pro-
file, and then taking an average over the 20 drops at each level.
The depth-averaged part of each profile was removed because it is
unknown and is not the true barotropic part of the velocity profile
(see Sanford (1971)). The resulting low-frequency profiles are
shown in Figures 4 and 5 for the east and north components.
There is some indication in these mean profiles of a decrease
in the mean shear in the 18* water (around 400 dbar). There is
also an indication that the mean shear increases again near the
bottom, in the Antarctic Bottom Water. In terms of the geostrophic
shear relation this might be expected. Figure 8 shows the mean
Brunt-Vaisal profile at the location of the time series. Since
the Brunt-Viisala frequency is proportional to the square root of
the vertical potential density gradient, this profile shows that
the vertical density gradient has a relative minimum at about
~4~ III_~YI__I^_______1II__)__ -~ll ~ l~ ~ -LI~ si-L
5- DA Y SERIES
AVERAGE EAST PROFILE
-30 -20 -10 01000 -
10001
2000
3000
4000
5000
6000
(CM/SEC)
20
Figure 4. Average profile of the east velocity componentduring the five-day time series.
30
__ L__~ rl__~_Y)___I-Y ~ I--1PII~YL~U
26.
5- DAY SERIES
AVERAGE NORTH PROFILE (CM/SEC)
-30 -20 -10 0 10 20 300 1 I 1 I
I I I I I I I I I
Average profile of the north velocity componentduring the five-day time series.
1000 -
2000 -
3000 -
4000 1
5000-
6000
Figure 5.
~--_-~-e-U-iiPYl~i ~- _III--.L.^PILYL1-II~ --LIIIL.. ~~~IL~-
400 dbar, and increases slowly from about 4500 dbar to the bottom.
The vertical shear of a geostrophic flow is proportional to the
horizontal density gradient. It is reasonable to expect that the
shear flow has caused the density field to distort in such a manner
that the horizontal density gradient at a point is proportional to
the vertical density gradient at the same point. Keeping in mind
the fact that we have assumed the low-frequency flow to be in geo-
strophic balance, the above would seem to be a reasonable explana-
tion of the observed decrease in shear of the mean flow at 400 dbar
and of the increase in this shear as the bottom is approached.
c.) High-frequency velocity components in the upper 2.5 km as afunction of time
From the 20 high-frequency profiles in the five-day time
series, (that is, the profiles obtained after the mean velocities
of Figures 4 and 5 have been removed), contour plots of east and
north velocity components have been made (Figures 6 and 7). These
plots show how various structures in the horizontal velocity field
evolve as time progresses. This evolution appears as changes in
the depth and size of individual features. Before discussing the
significance of these plots, a few comments on their construction
should be made.
First, the distribution of profiles in time is not uniform.
In particular, there are relatively large gaps in time between 224D
and 226D, and between 228D and 230D. Between these two pairs of
drops the profiler had to be used for inter-comparison drops with
CONTOURS OF ZERO EAST VELOCITY COMPONENT
TIME HOUR oTIME
DATE
DROP o
ONUMBER
500
lu
(I)
1000
1500
2000
2500
8.0
000012-VI-73
000013-VI-73
000014-VI-73
1000 0 0 00000
15-VI-73
( 0 N 3 a N 3 O O N 0 NO - NM to W I-M 0 t ) tonMM M n ) fn n ) qIICM (M w CQ< ~ ~ CY CY Cq( C CY y C
Figure 6. Contours of the east velocity component during thefive-day time series. Negative east components are
indicated by hatched regions.
~~Ll~i i ___ I _ -~--illl~WILLIX-- ICI. .-. -_ .~~-~-1L--IP~ LILC~PLY II.
CONTOURS OF ZERO NORTH
60oo00oo
13-VI-73
800000
14-VI-73
1000000
15-VI-73
Contours of the north velocity component during thefive-day time series. Negative north componentsare indicated by hatched regions.
TIME HOUR oDATE 0000
12-VI-73
DROPNUMBER
500
1000
1500
2000
2500Figure 7.
__
VELOC/TY COMPONENT
other vertical profilers, and these inter-comparison drops were not
done at the central mooring. Hence, gaps in the time series re-
sulted. Also, the time interval between profiles in the second
half of the series (after 230D) is generally less than it is for
profiles in the first half. This uneven time distribution allows
some regions of the plots to be contoured more easily than others.
Second, only the 0 cm/sec contour line is shown. Since the
main intent in drawing the contour plots was to attempt to see
vertical phase propagation of the near-inertial waves, a more de-
tailed contouring was not required. Regions with negative velocity
components are indicated by oblique hatching in Figures 6 and 7.
Finally, features for which contouring was ambiguous are
denoted by dashed-line contours. There are two main reasons for
this ambiguity. One is that as the depth increases, the overall
energy level of the profiles decreases (see Figure 8). Thus,
at depths of 2000 dbar to 2500 dbar, the true behavior of the
zero-crossing lines becomes clear. Below 2500 dbar it is very
difficult to draw contours with any confidence. The other reason
is that, even at shallow depths, certain profiles have regions
where a velocity component is close to zero without being clearly
positive or negative. In cases where it appeared that structures
on either side of this ambiguous region were clearly negative,
dashed-line contours were used to join these negative-velocity-
component features together. In other cases, there are regions on
one drop which are clearly positive, while adjacent drops have
II1)L-~ --IPI~PU~ 1\_CILIL--^IIXp_ _ I*CII
negative structures at the same depth. Such a case occurs at 500
dbar, around drop 242D in the contour plot for the north component.
In these cases, no attempt was made to join adjacent negative-vel-
ocity-component features. Taking note of these difficulties, we
can now point out some of the significant features in these plots.
Perhaps the most striking characteristic is the tendency for
regions of negative velocity to move upward in time. If we assume
that we can connect contours through regions where a velocity com-
ponent is not clearly positive or negative, then most of the struc-
tures with negative velocity can be followed across much of the 103
hours over which the time series lasted. If, on the other hand, we
assume that this connecting of structures is not valid, these nega-
tive-velocity regions will be divided into a series of "hot-dogs,"
with gaps of weakly positive velocity in between. In either case,
the upward motion is evident.
In a few cases, features do appear to move downward in time.
An example occurs at about 1000 dbar in the north plot, around drop
2280. It should be noted that this occurs in a region where the
profiles are widely spaced in time, and the contouring there is not
clear.
Several other points can be made concerning these two fig-
ures. First, if we follow the negative-velocity-component struc-
tures as they move upward in time, we see that, at a given depth,
it usually takes about a day for two successive negative-velocity
zones to pass through a given depth. This indicates that these
_I1__IDIYn____I__I__ ._ .-_lr---- -----..
structures have a period of about a day, which is near the local
inertial period at this latitude.
Second, if Figures 6 and 7 are overlaid, it is found that,
in general, negative-velocity-component zones in the east plots
occur at depths which are somewhat greater than the depths for the
corresponding zones in the north plots. In other words, the east
and north velocity components are not in-phase with respect to
depth.
Third, the speed with which the negative-velocity-component
zones move upward is about 2 mm/sec at 500 dbar, and this speed in-
creases to about 6 mm/sec at 2000 dbar. We will consider these ob-
servations in more detail in later chapters.
d.) The average perturbation kinetic energy profile, and a smoothedprofile of Brunt-V'isila frequency
The 20 high-frequency profiles (with vertical averages and
the low-frequency or "geostrophic" shear flow removed) were then
used to calculate profiles of high-frequency, or "perturbation,"
kinetic energy. This was done for each profile simply by squaring
the horizontal velocity values at each depth. These horizontal
kinetic energy profiles were then averaged, at each depth, over the
20 drops and a mean kinetic energy profile resulted. This profile
is shown in Figure 8. To simplify plotting, this mean, high-fre-
quency, horizontal kinetic energy profile was averaged in the ver-
tical by taking centered averages 50 dbar long around the points
50 dbar, 100 dbar, 150 dbar, and so on. Plotted along with the
-1--~ - -----I1'~1-~I^(-1~I*L.I1_II.-IIPL *~~I ii-_~_
MEAN PERTURBATION
KINETIC ENERGY (ERGS/CM 3)
0 I 2 3 4 5 6 7 8
N (CPH)Average profiles of high-frequency horizontalkinetic energy and Brunt-Vi1sa*l frequency.
1000
50
(()Z
u(.)
144(K'44
2000
3000
4000
5000
6000
Figure 8.
1PY-1L-~- -ltm... ~pn^-- II~~ .IXiL~LI~-- IC- .~PII*P-
ENERGYKINETIC
34
mean horizontal perturbation kinetic energy profile of Figure 8 is
a smoothed Brunt-Vaisala frequency profile, obtained from CTD (con-
ductivity-temperature-depth) stations made at roughly the same time
and location as the time series. Because of difficulties with
measuring velocities by EMVP near the surface (to depths of about
70 dbar), the points at 0 dbar and 50 dbar, showing rather high
kinetic energies, are probably not accurate.
It is possible to calculate an error bar for Figure 8. We
assume: the variance of the noise signal is 0.25 cm2/sec 2 for
either velocity component (Appendix C); noise at a given depth is
uncorrelated with the signal and is uncorrelated between profiles,
but the noise of points in the 50 dbar averages may be correlated
as may the noise in the east and north velocity components. The
contribution of the noise (if it is taussian) will then, for each
point in Figure 8, be distributed as a Chi-square random variable
with (at least) 19 degrees of freedom (Jenkins and Watts, 1969).
The result of applying the Chi-square distribution law is that the
difference between the measured and actual kinetic energy may be
expected to be between .18 and .43 erg/cm 3 95% of the time (the
measured kinetic energy being greater than the actual kinetic en-
ergy). This error estimate has not been plotted in Figure 8, since
it is rather difficult to see (it is about three times the width of
the line depicting the kinetic energy profile). We emphasize that
this error estimate only takes into account instrumental noise. It
does not take into account other sources of error, such as
35
contamination of the high-frequency profiles due to leakage of
high-frequency energy into the low-frequency profiles (Figures 4
and 5).
We see from Figure 8 that the profiles of mean horizontal
perturbation kinetic energy and mean Brunt-ViisilS frequency are
very similar. The horizontal kinetic energy decreases from the
surface down to about 400 dbar, increases in the main thermocline
to about 1000 dbar, and then steadily decreases down to about 4500
dbar. This parallels the behavior of the Brunt-Vais'li frequency.
It is interesting to note that while the Brunt-Vaisl frequency
slowly increases from about 4500 dbar to the bottom, the mean ki-
netic energy profile increases similarly to about 100-150 dbar off
the bottom, but then decreases again. This near-bottom decrease of
energy may be indicative of the presence of a bottom boundary layer.
The possible existence of a bottom boundary layer will be considered
in more detail in Chapter IV.
We recall that the time series lasted for only 103 hours.
This means that the mean kinetic energy profile was obtained by
averaging over approximately four wave cycles for the inertial
waves. It is not surprising, then, that there is considerable
structure in the mean kinetic energy profile. Presumably, if the
time series of profiles were of longer duration, the average per-
turbation horizontal kinetic energy profile would follow the mean
Brunt-Viisala profile more closely, particularly in the main
thermocline.
e.) The stretching of the vertical coordinate and the normaliza-tion of velocity by the mean Brunt-Vais3l8 profile
Figures 6 and 7 suggest that the high-frequency motions ob-
served are dominated by internal waves of near-inertial period
which are moving vertically through the water column. An important
quantity which could be calculated from the high-frequency profiles
is the horizontal kinetic energy spectrum as a function of vertical
wave number. However, the data on which we would like to perform
these calculations (that data being the ensemble of profiles in the
time series) is clearly a nonhomogeneous process. Figure 8 shows
that the overall variance (or horizontal perturbation kinetic en-
ergy) of the high-frequency profiles is a function of depth. Also,
an examination of Figure 3 shows that the most energetic waves have
vertical length scales which appear to be shorter in the thermo-
cline than in the deep water. This means that the statistical
properties of the wave field are functions of the Brunt-V~isal
frequency, which in turn is a function of pressure. Some method
must be found which will at least partially convert the process
represented by the ensemble of profiles into a homogeneous one.
Figure 8 gives an idea of the method to be used. Since the hori-
zontal kinetic energy tends to follow the mean Brunt-Vaisila pro-
file, we could make the variance more uniform with depth by normal-
izing the horizontal velocity at a given depth with the square root
of the Brunt-Vaisala* frequency at that depth. But this type of
normalization is what would be expected if the waves were obeying a
WKB approximation (Phillips, 1966), since in this approximation the
horizontal velocity components of an internal wave are proportional
to (N)1/2, where N is the local Brunt-Vaisla frequency. There-
fore, we chose to normalize the high-frequency velocity profiles
using:. - 1/2
u (z) = u(z)/(N(z)/N ) II-1
where u is an original velocity component, u is a normalized ve-
locity component (both functions of pressure, I), N(z) is the
Brunt-Vaisgli frequency at pressure z and N0 is a reference Brunt-
VMisMl frequency, o = 3 cph. No has been given this value to
facilitate comparison with theoretical vertical wave number spec-
tra presented later (see Chapter V).
We have also pointed out that the scale of the dominant
waves appears to change, being shorter in the main thermocline
than in the deep water. This is also in accord with the WKB ap-
proximation, which predicts that the vertical wave length of an
internal wave should be shorter in regions of greater N. Thus,
it is also possible to use the WKB approximation to normalize wave
lengths in the vertical. This has been done by stretching the
pressure coordinate according to the following differential law:
A= ( Az 11-2N0
where z is the original pressure coordinate, and z is the stretched
pressure coordinate.
Figure 9 gives z as a function of z, where z is in decibars
and z is in "stretched decibars." Below about 2000 dbar,i(z*) is
approximately a straight line. This, along with the fact that the
high-frequency perturbation kinetic energy is roughly constant with
depth below 2000 dbar, indicates that vertical wave number spectra
of the profiles below 2000 dbar could be approximately calculated
without use of the above normalization and stretching procedures.
But to include the thermocline in these calculations requires the
use of the above procedures.
Figures 10 and 11 show an example of a profile which has had
the above procedures (equations II-1 and II-2) applied to it. Fig-
ures 10 and 11(a) show the velocity profile 219D as it was origi-
nally obtained. Figures 10 and 11(b) show the same profile after
the "geostrophic" shear flow has been removed. Finally, Figures 10
and 11(c) show 219D after the above procedures for stretching and
normalizing the profile have been used. With the exception of
values in the top 100-200 stretched decibars, these profiles pre-
sent a much more uniform appearance than do the profiles of the
east and north components of 219D before stretching and normalizing.
The departure from uniformity in the top 200 stretched decibars
could be explained either by the fact that the WKB approximation
breaks down near the surface, where N is changing rapidly, or by
the fact that the shear of the mean horizontal velocity increases
near the surface. This shear is not taken into account by the above
WKB normalization.
PRESSURE (STRETCHED DEC/BARS)
I000
oi 2000
3000
q, 4000
5000
6000
Figure 9. Stretched and unstretched verticalpressure coordinates.
EA T (CM/) EA S T (NCM I/S
) -15 -10 -5 0 5 10 15 20 -10 -5 0 5
400-
,800-
1200-
-1600 -
20001
C
Figure 10. East component of profile 219D as originally obtained (a), after removal of themean profile (b), and after stretching and normalizing (c).
-20 -1
1000
2000
3000
5000
Lu154
(Zi44
a b
NORTH, (CM/S) NORTH, (NCM/S)
Figure 11. North component of profile 219D as originally obtained (a), after removal of themean profile (b), and after stretching and normalizing (c).
(i)
tj
LU
J
f.) Spectral decomposition of stretched profiles
The time series of profiles is now in a form (the stretched
and normalized profiles, such as 219D in Figures 10 and 11(c)) in
which the vertical wave number spectrum of horizontal high-frequen-
cy kinetic energy can be calculated. We now describe and present
several different types of spectra which have been obtained.
The first type, shown in Figure 12, is the simple autospec-
trum, or the total energy of the east and north components as a
function of stretched vertical wave number. For this spectrum, a
subset of nine down profiles out of the original 20 was selected.
The only reason for this selection was that some profiles have
better data near the surface than others. The EMVP quite often
does not obtain accurate velocity values for the first 50-70 dbar
below the surface. Since the WKB procedure described above tends
to weight values near the surface more heavily than val.ues in
deeper water, it was important to choose those stretched profiles
which have the best data near the surface. Thus, the best nine
down profiles were chosen for the calculations. The top 50 dbar in
the stretched profiles were also ignored when calculating the spec-
trum. This further reduced the influence of near-surface measure-
ment errors. The 95% confidence limits were calculated using an
assumed 36 degrees of freedom (Jenkins and Watts, 1968). Each
final spectral estimate is formed by averaging the original esti-
mates over four adjacent wave number bands in each profile, and then
averaging over the nine profiles. This would give 72 degrees of
WAVE LENGTH(SOB)
100
0.01
VER TICA L WA VE NUMBER(C/SDB)
Total spectrum (east and north energy) for 9 down profiles.
LQK
'4
(%)
cN4
102
20
0.0510
Figure 12.
freedom to each estimate, if east and north velocities are statis-
tically independent. However, Figure 13 indicates that the east
and north velocity components are not independent. Therefore, 36
degrees of freedom appears to be more realistic.
The somewhat unusual units in the spectrum of Figure 13 arise
because of the normalization by N(z). Thus NCM/S means "normalized
cm/sec," and C/SDB means "cycles per stretched decibar." If m and
m' are vertical wave numbers in the unstretched and stretched coor-
dinates, respectively, then the relation between them is
m = ( ) m' . 11-3
Normalized cm/sec are the units obtained when equation II-1 is ap-
plied to an original velocity profile.
The most interesting characteristic of the autospectrum is
that at high wave numbers it has a slope of about ,2.5 on a log-log
plot. At smaller vertical wave numbers this slope decreases. This
spectral shape will be compared in Chapter V with a theoretical
vertical wave number spectrum derived by Garrett and Munk (1972,
1975).
The second type of spectrum which is presented is a form
which treats the horizontal velocity vector (u,v) as a complex vec-
tor, u + iv, (Gonnella, 1972; Mooers, 1973). At any vertical wave
number, m', the Fourier transform of u and v separately gives sinu-
soids for each component. These two sinusoids, taken together,
will form an ellipse in the u, v plane, the position of the velocity
WAVE LENGTH(SDB)
100
0.01
20
0.05
VER TICAL WA VENUMBER(C/SDB)
Figure 13. Clockwise (CW)down profiles;
and counterclockwise (ACW) spectra for 9In the text, CW is C(m') and ACW is A(m').
103
C)
c
144
X 1(%)
101
10
vector on the ellipse being a function of z , the pressure in
stretched decibars (increasing downward). Since any ellipse can
be represented as the sum of two complex vectors rotating in op-
posite directions, but with the same frequency, the complex vector
umr + ivm, can be represented by
im'z* -im'tvum + iv, = u e + u e II-4
where m' is always considered to be positive, and u+ and u_ are
complex. The energy spectrum is then given as a decomposition be-
tween the clockwise energy, C(m'), and the counterclockwise energy,
A(m'), defined by:
C(m') = 1/2 u , u > ; II-5
A(m') = 1/2 < u+ u+ > , II-6
where the brackets denote an average over realizations (in this
case, profiles) and possibly over m', the stretched vertical wave
number. The total energy, T(m'), is given by
T(m') = A(m') + C(m'). 11-7
T(m') is plotted in Figure 12, and A(m') and C(m') for the same
nine drops are plotted in Figure 13.
Probably the most striking aspect of Figure 13 is the fact
that clockwise energy is greater than counterclockwise energy over
much of the stretched vertical wave number range. Only in the high
wave number region of the spectrum, where the energy is relatively
low, do we find points where counterclockwise is greater than
clockwise energy. This means that the waves are elliptically
polarized in the clockwise sense, or in other words, that there is
a distinct tendency for the horizontal current vector to rotate in
a clockwise sense with depth, as seen by an observer looking down
from above. This has important implications for the theory of
internal waves, which will be discussed in the next chapter.
The third type of spectrum that we present here is simply
that obtained by plotting the energy of the east and north compo-
nents separately. This spectrum is given in Figure 14 for the
same nine down drops as in Figure 12. We see that, in contrast to
the difference in Figure 13 between the clockwise and counterclock-
wise spectra, Figure 14 shows that there is almost no difference
between the energy of the east and north velocity components. This
is an indication of isotropy; that is, that in the mean there is as
much energy in the east-west as in the north-south direction.
Finally, we present in this section results of spectral cal-
culations performed on nine up profiles from the same time series.
It often happens that on a single drop (up and down) the down part
will have good near-surface data while the up part will not. For
this reason, the nine best up profiles chosen do not coincide with
the nine best down profiles. These spectra are shown in Figures
15, 16 and 17. Each spectral estimate represents an average over
four adjacent wave number bands and over nine up profiles. The
major reason for performing the same calculations for up and down
WAVE LENGTH(SDB)
100 20
0.01 0.05
VER T/CA L
Figure 14.
WA VE NUMBER(C/SDB).
Spectra of east and north velocitycomponents for 9 down profiles.
K,
102
I0'
C)
XP
10
WAVE LENGTH(SDB)
100
0.01
Figure 15.
VER 7TCAL -WA VE NUMBER(C/SDB)
Total spectrum (east and north energy) for 9 up profiles.
LLJ
LU
Q:
czK1
X
C\4
10"
102
20
0.0510
WAVE LENGlTH(fSDB)
100
0.01
20
0.05
VER 7TCAL WAVE NUMBER(C/SDB)
Figure 16. Clockwise (CW) and counterclockwise (ACW) spectra for9 up profiles.
k
C~j
(%)
K-
102
10'
X0
C\
100
WAVE LENGTH(SDB)
100 20
0.01 0.05
VER TI CA L WA VE NUMBER(C/SDB)
Figure 17. Spectra of east and north velocitycomponents for 9 up profiles.
103
102
K
LU
(%)I:::l4~J
C
(I10
10
profiles was to show that there is very little difference between
them (even though all of the same profiles were not used in both
calculations). That the two computations are very similar can be
seen from comparing Figures 12 and 15, 13 and 16, and 14 and 17.
The 95% confidence limits (with 36 degrees of freedom) have
been included for the total vertical wave number spectra (Figures
12 and 15) and for the spectra of east and north velocity compo-
nents (Figures 14 and 17). These confidence limits should be used
with caution, however, since they strictly apply only to a white
noise spectrum. The spectra presented in this section are clearly
"red." Even though the original profiles were pre-whitened (by
taking first differences) before applying the 7ourier transform,
and the above spectra were then produced by re-coloring the spectra
obtained from the first-difference pfofiles, it is likely that the
pre-whitening did not completely convert the profiles into ones
with flat spectra. This means that the actual number of degrees
of freedom could be less than 36. We have also not plotted confi-
dence limits for the clockwise and counterclockwise spectra.
The clockwise and counterclockwise energy estimates depend not only
on the probability distribution of the autospectra for the east and
north velocity components, but also on the probability distribution
of the quadrature spectrum between east and north velocity compo-
nents (Appendix D). Instead of deriving a theoretical estimate for
the probability distribution of the clockwise and counterclockwise
spectra, it is probably more important to point out that, as
LILILYl~__laYIJI__lr^..._C Y-III~LIIII I .~-l._)i- l__ ~^__.~--~.--ri--il1liL
indicated below, clockwise and counterclockwise spectra calculated
from four-drop groups in the time series always showed that clock-
wise energy was greater than counterclockwise energy over most of
the stretched vertical wave number range. Furthermore, while av-
eraging the spectra of two four-drop groups together gave clockwise
and counterclockwise spectra very similar to those of Figures 13
or 16, averaging three four-drop groups together did not appreciably
change the spectra. This indicates that nine profiles are enough
to give stable estimates of the clockwise and counterclockwise
spectra.
Before the spectrum of Figure 12 was calculated, five pre-
liminary autospectra were obtained by using four-drop groups in
the series of 20 down profiles. That is, drops 219D-223D were used
to obtain a spectrum, then drops 224D-228D, and so on. All four-
drop groups showed a dominance of clockwise over counterclockwise
energy. Each of the five preliminary spectra also showed a tend-
ency for the energy to peak at vertical wave lengths of 100-130
sdb. Drops 230D-233D, in particular, showed a strong energy peak
there. However, in some cases (for example, 219D-223D) this peak
did not show up as clearly. The fact that the energy spectra did
appear to have some dependence on time indicates that the spectra
calculated from the nine best profiles out of the 20 will tend to
smooth out peaks in the energy spectra.
On the other hand, it is clear that energy contributed at
vertical scales of 100-130 sdb is an important part of the stretched
I~IUlrl_ i L~*~l_~__l~l~ _ ~I___ _ 1.111~---
profiles. From Figure 3 we see that the most apparent visual var-
iability in this profile occurs in the above range of vertical
scales. (In the original pressure coordinate, the above scales
correspond to 100-150 dbar in the main thermocline, and reach
1000 dbar in the deep water.) A comparison of drops 219D and 2210
in the stretched vertical coordinate also shows that much of the
"mirror-imaging" that occurs between these profiles is caused by
waves with the above vertical length scales.
g.) Paired profiles
During the course of the time series experiment, several
simultaneous paired profiles were made, using two EMVP's. These
were done at separations of from 100 meters to about 15 kilometers.
Figure 18 shows two profiles which were made simultaneously at a
separation of 4.8 kilometers. Profile 235U was made to the south
of profile 236U. It can be seen from this figure that many velocity
features of 235U can be traced visually to corresponding features
in profile 236U, although there is quite often a depth change be-
tween the more energetic features of profiles 235U and 236U. This
indicates that: a.) many of the features observed in the two
series (at least those with wave lengths greater than about 100
dbar) have a large ratio of horizontal to vertical length scales;
and b.) the energetic features in the two profiles are not hori-
zontal, but are slightly inclined. We will return later to a con-
sideration of the inclination of velocity features between 235U and
VELOCITY PROFILESSIMULTANEOUS
235U & 236UDROPS 4.8 KM APART
EAS7 (CM/S) NORTH, -. (CM/S)
()
Figure 18. Simultaneous paired drops, 235U and 236U, separatedhorizontally by 4.8 km.
56
236U. We point out here, however, that waves with large ratios of
horizontal to vertical wave lengths should, according to the re-
sults of Appendix B, be measured with small error by the EMVP.
Chapter III Vertical Polarization and the Vertical Propagation of
Internal-inertial Waves
a.) Summary of data
Before going further, it will be useful to summarize the
results of the previous chapter. Figure 3 indicates that high-
frequency energy in the profiles is dominated by waves near the
local inertial frequency. There is probably also energy contri-
buted by diurnal and semidiurnal tides. Figures 6 and 7 show that
the velocity structures associated with these inertial waves
clearly move upward with time at the time series location. These
two figures also show that the east and north components of hori-
zontal velocity are usually not in-phase with depth, but that the
east component lags behind the north component. This indicates
that these near-inertial waves are polarized with depth*, a fact
which is born out by the clockwise and counterclockwise spectra of
Figures 13 and 16. These spectra show that the waves are ellipti-
cally polarized with depth, and they show that this polarization
is predominantly clockwise (the clockwise energy is greater than
the counterclockwise energy over most of the observed wave number
band). We note that the observed lag in Figures 6 and 7 between
the east and north components also indicates that these waves are
polarized clockwise with increasing depth. Finally, Figures 14 and
17 indicate that there is very little difference between the energy
of the east component and that of the north component. We would now
like to see how the above observations may be related to internal
wave theory.
b.) The vertical spatial behavior of horizontal internal wavevelocity components
Expressions for the east and north, or u and v velocity
components of a single propagating internal wave are given in the
WKB approximation by:
S= ( ak i ifl) x A (N2 () _ 2) 1/4
v ( + ikf T wl) ) (2 _ f2)l/ 2 (k2 + 12)1/2
k2 + 121/2 1/2+ i(- (N2() - 2) dzS 2 f2 (N i(kx + ly - wt)
xe x eIII-1
where A is the wave amplitude, and tie coordinate system is chosen
so that x is positive eastward, y is positive northward and z is
positive upward. The coordinates 2 and 2* will be reserved for the
vertical pressure coordinate (increasing downward) in decibars and
stretched decibars, respectively. (The reader may refer to Chapter
V for the derivation of these formulas.) In order to simplify
these expressions, we rotate the coordinate system through an angle
a as shown in Figure 19. In the new coordinate system:
0' = 1 cosa + 9 sina ;
v' = - sina + v cosa ;
k' = k cosa + 1 sina ;
l' = -k sina + i cosa = 0 , 111-2
44,9
t
v I /"0 ,'%
u KX
Figure 19. Horizontal coordinate rotation (see equations 111-2).
-h
.4-,
I
Figure 20. Direction of rotation of the horizontal velocity vector of aninternal wave for positive and negative vertical wave numbers.The curved arrow shows the direction in which v rotates withincreasing z.
~LI~I iliuuruu~--l*i~-*~r+-p- ~~...~_li--r-------cI~ -- ~P--L~- ---~Y__
ii'
IKr
where
COSo = kik 2 + 12
and
sina =/ k2 + 12
In the new coordinate system, we have
u' = 1 xA (N(z) 2 /4
S f 112(z) /2)+ i- (W2 _ f21/2
+ i f2) 1 2 f(N2(z) _W2) 12
z i(k'x' - at)x e
k 2 1/2k' 2 2
2 /2- 2) dz Z m(z) z , III-5
where m(z), the local vertical wave number, is given approximately
by
m(z) 2= k 22)1/2 (N2(z) - 2)/2
Note that m(z) is a positive quantity. The choice of signs in the
expressions for horizontal velocity determines whether the wave is
111-3
x e
We let
SIII-4
111-6
propagating upward or downward. Then
(N2 2) 1/4 i(k'x' + mz - wt)u' = l (N2(z) -W )9' =+ --f (2 - f2)/2 -7
- {O
The horizontal component of the wave number vector, k', is assumed
to be in the direction of horizontal phase propagation, so w > 0.
We note in passing that i' and ' are proportional to N1/ 2, while
m(z) is proportional to N. This confirms the connection made in
Chapter II, without proof, between the WKB approximation and the
procedures used for stretching and normalizing the velocity pro-
files. Taking real parts of 5' and 9':
Re (5') = u' = + cos (k'x' + mz - t) xA (2(z) - 2) 1/4
Re (9') = v' = -f sin (k'x' + mz - wt)S (wZ2 f 2)/ 2((W
III-8
The upper signs are chosen if m is directed upward, and the lower
signs are chosen if m is directed downward. A is assumed to be
real. To observe the behavior of u' and v' as z varies, we set
x' = t = 0. Then
1/4u' = F cos (+ mz) xA (N2(z) - ) 1/4
v' = - sin (( mz) (2 f2)/ 2 III-9
The two cases for the choice of upper or lower signs are shown in
K
*Cg
C9
KFigure 21. Propagation diagram for internal waves. Cg is the group
velocity vector, k is the wave number vector, and v is avector representing water velocity.
_II__IWQYJIP_____III- Il~i--X- - - ~
4. 4 2 1/4Figure 20. We note that ' V'/(N2(z) - . The horizontal
4.vector, v', is plotted in Figure 20 instead of v' in order to re-
move variations in the horizontal velocity vector magnitude brought
about by changes in N(z). The ellipses represent hodographs of the
horizontal velocity vector as z varies. As w -+ f, the wave becomes
circularly polarized. We note that although we have been writing
simply A for the wave amplitude in the above expressions, the wave
amplitude is more generally a function of vertical wave number,
frequency, and a, the direction of horizontal phase propagation;
that is, A = A(m, w, a). In the + m case, the horizontal velocity
vector rotates counterclockwise with increasing z, hile in the - m
case, the sense if rotation is the opposite. The result is that
there is a unique relation between the sign of m and the sense of
polarization in the internal wave hodographs of Figure 20.
We can now relate the sign of m(z) to the direction of the
vertical'group velocity component for internal waves. Figure 21
shows a propagation diagram for internal waves. The vector t is
the wave number vector for the wave and ig is the group velocity
vector (or the direction of energy propagation).
For simplicity, only two wave number vectors have been
shown, one pointing toward the sea surface (toward positive z) and
the other pointing away from the sea surface: the dispersion rela-
tion gives
W2 = N2 sin2 (0) + f 2 COS 2 (0), III-10
I_ _
644.
where 0 is the angle that the wave number vector, k, makes with
the z axis. For w close to f, 0 is small, and t is almost verti-
cal. For constant N, i is also independent of depth. On the other
hand, if N varies with depth, then e and the vertical component, m,
of k also change with depth. If the wave propagation may be approx-
imated by a WKB expansion, however, equations III-10 and III-11 re-
main true locally, except that m and N are now functions of z. The
magnitude of the group velocity is given by:
Cg 1 a : (N2 _ f2) sin-0 cos 0
I'l a IIl [N2sin20 + f2cos2 O]1/2 III-11
The group velocity vector is perpendicular to i and points away
from the z axis if N > f, which it was everywhere in the water
column during the time series experiment. The vector v in Figure
21 is the water velocity due to the wave. Since the water is as-
sumed to be incompressible, i v = 0 always. There is no depen-
dence on horizontal direction in the above formulas, so the t and
tg vectors can be rotated about the z axis without change of fre-
quency or magnitude of tg. We have also drawn two helices in
Figure 21. These helices represent the path that the tip of the
velocity vector, v, makes as distance from the origin is increased
along k.
Looking down on this figure from above the z axis, it is
readily seen that the sense of polarization of the wave (that is,
the direction in which v rotates with increasing depth) changes,
654.
depending on whether k is directed toward or away from the sea sur-
face. For a vector k directed toward the 5ea surface, the wave is
polarized clockwise with increasing depth, while for a k vector
directed away from the sea surface, the wave is polarized counter-
clockwise with increasing depth. This is simply a restatement of
the result expressed by equations III-8 and III-9.
We point out that polarization can be inferred easily from
the behavior of internal waves in time. We assume we are observing
at a fixed point within the ocean in the Northern Hemisphere. Then
it is known that a velocity vector due to an internal wave propa-
gating through that point will rotate in a clockwise sense with
time, due to the influence of the Coriolis acceleration. Since
the internal wave is assumed to be propagating along the direction
of the wave number vector, t, of Figure 21 (either toward or away
from the sea surface), the waves have to be polarized as shown in-.
the figure in order to give a velocity vector, v, which rotates
clockwise in time at a given point in space.
Now, Figure 21 also shows that if t is pointed toward the
sea surface, Cg is pointed away from it, and vice versa. There-
fore, if either the sense of wave polarization or the direction of
the vertical phase propagation (the sign of m) can be determined,
then the direction of the vertical component of the group velocity
vector, and thus the direction of vertical energy propagation, can
be found (Leaman and Sanford, 1975).
The data presented in Chapter II allows us to determine both4.
the direction of the vertical component of k (at least for the
dominant waves) and the sense of wave polarization in the vertical.
As already pointed out, the dominant waves in Figures 6 and 7 al-
most always appear to be propagating upward through the upper 2.5
kilometers of the water column. This indicates that the direction
of phase propagation, or i, for these waves is pointed toward the
sea surface. If this is true, the above discussion shows that the
waves should be polarized predominantly clockwise with increasing
depth. Both the clockwise and counterclockwise spectra of Figures
13 and 16, and the fact that the east and north velocity components
are out-of-phase in depth, with north leading, in Figures 6 and 7,
show that the waves are in fact dominated by clockwise polarization.
The influence of the mean flow on the observed waves has so
far been ignored. The depth dependent part of the mean flow pro-
file has already been presented (Figures 4 and 5). Another depth
independent component of the low-frequency profile may also exist,
but its magnitude cannot be evaluated from EMVP data (Sanford,
1971). In order to evaluate a possible depth independent part of
the mean profile, we have calculated the average horizontal cur-
rent at the central mooring as measured by a current meter at
1513 dbar (1496 meters). The average horizontal velocity compo-
nents over the interval of the profiler time series, as determined
from the current meter record, are + 1.6 cm/sec to the west and
+ 2.5 cm/sec to the north. The magnitudes of the horizontal current
- - , - 0, - , , 6 .6 'dM --- - - , __ __ -
components at 1513 dbar in Figures 4 and 5 are ~ 1.5 cm/sec to the
west and ~ 1.5 cm/sec to the north. This indicates that there is
very little shift in the east component, while - + 1 cm/sec should
be added to the north profile to make the average north flow ob-
served by the current meter and the profiler agree.
Since the wave number vector, t, is not exactly vertical
for waves propagating in the vertical, we expect that the advection
of waves by the mean flow will cause an apparent vertical propaga-
tion. The amount and sign of this apparent propagation (upward or
downward) depends on the direction of horizontal wave propagation
relative to the mean flow. Figures 4 and 5 show that the main part
of the mean flow is oriented north-south. Thus, for example, if
the waves were propagating east-west, there would be little appar-
ent vertical propagation due to the advection of waves past the
observing point. On the other hand, if most of the waves were
propagating northward (and upward), the advection would cause an
apparent downward phase propagation to be added to the true upward
phase propagation of the wave above 1500 dbar. In the absence of
adequate information on the direction of horizontal phase propaga-
tion it is difficult to estimate the influence of the mean flow.
There is some theoretical support for assuming that the near-iner-
tial waves are propagating northward (Munk and Phillips, 1968).
If this is, in fact, true, Figures 6 and 7 show that k must be di-
rected toward the sea surface: that is, the actual upward phase
propagation of the waves is enough to overcome any apparent
downward propagation caused by advection. A strong indication
that the observed waves are propagating upward can be found by ex-
amining Figures 4 and 5. It is known from tracking neutrally
buoyant floats that the mean, or "low-frequency" flow is small at
depths of around 1500 dbar (Voorhis, 1974). Figures 4 and 5 also
show that, taking into account the shift in the north profile re-
quired to make the mean flow as observed by the profiler and cur-
rent meter equal at 1513 dbar, the mean flow in the depth range
1400-1500 dbar is small (< 3 cm/sec). Therefore, there should be
little advection of waves at this depth. However, Figures 6 and 7
show that the waves are moving upward at this depth. In fact, the
rate of upward motion is greater at 1500 dbar than it is at lesser
depths. Vle note that the sense of polarization is not affected by
the advection of waves past the observation point. Therefore, for
the purpose of determining the direction of vertical wave energy
flux, observation of polarization appears to be more useful than
observation of the apparent vertical phase propagation.
The behavior of wave polarization and the direction of
phase propagation both appear to indicate that the vertical energy
flux is directed downward, away from the sea surface. It is nec-
essary, however, to discuss just what the partition of energy be-
tween clockwise and counterclockwise means for these waves. In
particular, we have limited the above discussion to a single wave.
It is more appropriate to relate the partition of clockwise and
L -LLillllslC -~d~~^---CI" IPt-~ -C-~~^~ I
69
counterclockwise energy to the energy carried by upward and downward
propagating waves. This will be done in Chapter IV.
It is known from internal wave theory (see, for example,
Garrett and ,unk, 1972) that internal waves with frequencies near
the inertial frequency are isotropic in time; that is, the magni-
tude of the frequency spectrum of a near-inertial wave horizontal
velocity component will be independent of the direction of that
velocity component. The same isotropy will be true for the verti-
cal wave number spectra of near-inertial waves, provided only that
east and north spectra for a sufficient number of profiles are
averaged together.
To see why, this is true, we consider a single internal wave
propagating energy downward. The u' and v' velocity components are
given by equations 111-8:
uL = - cos (k'x' + mz - wt) (N2 - )1/4
V1 - sin (k'x' + mz - wt)~ (2 - f2) 11 2 111-12
where A is the wave amplitude. -We-point-ot- that the hodograph in
z for this wave is given by:
u' 2 2 (N2 - W2)1/2
1 + a A2(m,,)( )2 ( 2 - ) 11-13
At any time, the ratio of semi-major to semi-minor axis is - , andf
as w - f, the wave becomes circularly polarized. If we assume the
presence of a second wave propagating upward with the same vertical
wave number magnitude, m, and the same frequency, w, but with dif-
ferent amplitude, B, then the velocity components for this wave are
given by:
u= cos (k'x' - mz - wt) x1/4u x (N2 .)2)
vu f sin (k'x' - mz - t) (2 f2) 1/2 I-14W - 111-14
We assume A and B to be real now, although this is not necessary.
We also assume we are observing at the coordinate origin, so
x' = 0. The result of adding the two waves together gives:
1/4S (N2 )2
UT= (/2 [(B - A) cos(mz)cos(wt) - (A + B) sin(mz)sin(wt)],
(W2 - f2)
1/4
v' = (N - ) [(A - B) cos(mz)sin(wt) - (A + B) sin(mz)cos(wt)].(W2 - f2) /2
If-15
Weisee that, for a fixed z, the magnitude of u' averaged
over a cycle in time, becomes equal to the average of v' as w - f
independently of the magnitudes of A and B. Thus, spectra of u'
and v' in frequency may be expected to show the same energy when
the frequency is close to f.
The same is not true, however, if vertical wave number spec-
tra of u. and v are considered, as can be seen by fixing t in the
above expressions. If A or B is zero, the vertical wave number
spectra for u' and v' will be approximately equal for w close to f.-r
IIIIIL_-LII~I~_.~ ~_ -~.--.I~ ~
However, if A = B (which would occur if standing waves existed in
the vertical), energy in the vertical wave number spectra of u' and
v.' will depend on t, the time in the wave cycle at which the spectra
were obtained. For example, in this case, at t = 0, the spectrum
of v" will contain all the energy at that particular vertical wave
number, m, and the spectrum of u' would, in theory, be zero.
However, if vertical wave number spectra of u' and v' are_r _
taken at enough different times, t, in the wave cycle, we again ap-
proach isotropy for w -+ f. This is true because, at a given m, the
energy in u' and v' will be the same if averaged over time. Ther _T
average vertical wave number spectra of Figures 14 and 17 can be
considered as the sum of individual wave numbe- spectra taken at
numerous different times, t, in a wave cycle. Then the fact that
much of the energy in the profiles apears to be contributed by
waves with frequencies close to f explains why there is an equi-
partition of energy between the east and north components.
c.) Polarization in the deep water
As mentioned previously, we can approximately calculate deep
water spectra (below 2500 dbar) without stretching and normalizing
the velocity profiles. This is possible because N does not change
much in the deep water. Calculating the clockwise and counter-
clockwise spectra over the depth range 2500-5500 dbar for several
profiles indicates that dominance of clockwise energy over counter-
clockwise energy exists in the bottom half of the water column, at
least for vertical wave lengths greater than about 100 dbar. This
-i--L~1' ~--sur~ u*rp*_^-,r-r~i.~.... -~-I*--.--- ~ ̂-^. ~----- i ---*- e~-rr.
is important because it indicates that, even in deep water, the
net flux of energy by these near-inertial waves is downward.
Pollard (1970) indicated in a theoretical study that it would be
difficult for near-inertial waves to propagate down into the deep
water with the observed intensity. The result of the above cal-
culations indicates that these waves are, nevertheless, coming down
from regions farther up in the water column. The Pollard model
assumes that inertial waves are generated at the surface by an im-
pulsive force that acts over some time and space interval. In
Chapter V we will examine the behavior of the vertical energy flux
of near-inertial waves for an assumed form of the internal wave
energy spectrum in frequency and vertical wave number space.
d.) Comparison with meteorological observations
As was pointed out in the introduction, meteorologists have
been conducting experiments to obtain vertical profiles of atmos-
pheric winds for many years. Their results may be compared with
results obtained by the EtMVP in the ocean.
Various authors have pointed out that the existence of in-
ternal waves in the atmosphere has not been proved. For example,
Justus and Woodrum (1973) felt it necessary to state that: "There
has been no unambiguous resolution of gravity waves in the upper
atmosphere (i.e. simultaneous observations of amplitudes, phases,
and frequency sufficient to verify propagation according to the
theoretical dispersion equation)." They go on to state that there
IIU____CIYI___I___~_i~-C~ LIII_ i I~l _. ~X ~~-~ I~IUYL~
is, however, strong circumstantial evidence for the occurrence of
internal waves. Some of that evidence will now be presented.
In 1966 radar tracking of radar-reflective balloons was used
at Cape Canaveral to obtain profiles of atmospheric winds from
0 kilometers to about 16 kilometers (Endlich, et. al., 1969). Two
series of profiles were made, each series lasting between one and
two days. The method allows one to sample rather rapidly in the
vertical (one point each 25 meters) but, unfortunately, it can
only determine the wind speed at each point, not its direction.
The authors point out that there are strong velocity features in
the July 4-5, 1966, experiment which progressively move downward
with time. The profiles are complicated, however, and no definite
time or space scales stand out (although it is known that internal
waves can exist in the atmosphere at these altitudes).
Although apparent high-frequency features in the atmosphere
have been observed to change altitude, there appear to have been
few reports of any polarization or directional structure associated
with these features. Endlich could only measure speed as a func-
tion of altitude and therefore could not observe polarization.
Hines (1966) has reported polarization of atmospheric winds that is
apparently related to the diurnal atmospheric tide. His data, ob-
tained by tracking sodium vapor trails at heights of 90-130 kilo-
meters, indicates that winds with periods appropriate to the diurnal
tide rotate with altitude in a given rocket profile. If an observer
is imagined to be standing above the atmosphere and looking down,
II~L1 YI_~ I_~L _ CJI^^___I__IY__I_~ PYY-
then the waves seen by Hines have horizontal velocity vectors that
rotate counterclockwise with increasing "depth," or atmospheric
pressure. This is the opposite of the case observed in the ocean
by the EMVP over the smooth topography, in which the horizontal
current vector rotates clockwise with increasing depth. As Hines
points out, the atmospheric observations indicate that the wave
energy of these oscillations is propagating from lower altitudes up
to the 90-130 kilometers range. This appears to be one of the few
cases where wave polarization has been observed in the atmosphere
(C. 0. Hines, 1975, personal communication).
One good reason for a lack of elliptical or circular polar-
ization in the motions interpreted as being internal waves in the
atmosphere is the fact that these motions quite often have periods
much shorter than the local inertial period. Hines (1960) points
out that atmospheric internal waves might be expected to be most
energetic at periods of several hours. For waves of this period,
the earth's rotation is not a major influence, and a single atmos-
pheric internal wave with a three hour period would be almost
linearly polarized in the vertical.
Hines also notes that some observations in the atmosphere
show structures which progressively move downward in time. Using
the same arguments as those put forth in describing Figure 21, he
points out that because the directions of vertical phase propaga-
tion and energy flux are opposite for internal waves, these down-
ward moving structures in the upper atmosphere could be interpreted
I~ I^IC~Llli-O-IWI I(C- --U1 X~I-l~~llll^n-^~ YTrIIYYIII XI-~l~-l-- ~~ --LI_-lllll~l llll
as internal waves carrying energy upward, away from the lower at-
mosphere and the surface of the earth. (The sources for atmospheric
internal waves are thought by many meteorologists to be located in
the lower atmosphere or at the earth's surface.)
Finally, there is some evidence from fresh water lakes that
appears to indicate upward phase propagation of internal waves.
Lazier (1973) obtained a series of temperature profiles in Lake
Bala, Wales, over a time interval of about 12 hours. Profiles were
repeated at approximately 11 minute intervals. In many cases,
features in the temperature profile were observed to move progres-
sively upward in time. Lazier attributes this upward motion of
temperature featt.res to the presence of propagating internal waves.
If the energy source for these waves was located at the lake sur-
face, they would have to propagate their energy downward. Then,
as in Figure 21, the wave number vectors of the waves wbuld be
directed toward the lake surface, and the wave phase would move
upward in time. Since Lazier only measured temperature profiles,
however, no data on wave velocity polarization were obtained.
Chapter IV The Use of the Ratio of Clockwise to Counterclockwise
Energy as a Reflection Coefficient. Reflection of Internal Waves
From a Smooth Bottom
a.) Calculation of a reflection coefficient for the observed waves
We pointed out in Chapter III that the dominance of clock-
wise over counterclockwise vertical wave number spectra for hori-
zontal kinetic energy appears to indicate that the net vertical
energy flux for these waves is downward. However, a further step
must be taken if we are to relate the partition between clockwise
and counterclockwise energy to the energy in upward'and downward
propagating waves. This step is to assume that the observed waves
have frequencies close enough to f so that any single wave should
be almost circularly polarized in the vertical (see equation 111-13).
The sense of polarization is determined by whether the wave energy
flux is downward (clockwise polarization) or upward (counterclock-
wise polarization). With this assumption, the horizontal kinetic
energy of the downward propagating waves is approximately equal to
C(m'), the clockwise energy in the vertical wave number spectrum,
while the horizontal kinetic energy of the upward propagating waves
is approximately equal to A(m'), the counterclockwise energy in the
same spectrum. (Strictly speaking, A(m') and C(m') should be multi-
plied by the Brunt-Vaisila frequency to get the true energy level
(see Chapter V). However, since we are finally interested only in
~YY~ ?~l_~~ _il~LI~- ~------C
the ratio of clockwise to counterclockwise energy, this step has
been ignored.)
We have assumed in the previous paragraph that inertial
waves are the primary cause of high-frequency energy in the pro-
files. However, frequency spectra of horizontal velocity obtained
at the central mooring (M. Briscoe, 1975, personal communication)
indicate that the diurnal and semidiurnal tides also contribute
to the internal wave frequency range.(see Figure 29). Because of
the relatively short length of the time series, it is difficult to
resolve semidiurnal tidal and inertial oscillations in frequency,
and it is impossible to resolve diurnal tidal and inertial oscilla-
tions. On the other hand, we can make inferences concerning the
probable influence of the tides on the vertical wave number spectra
computed in Chapter II. First, that part of the tidal energy which
is caused by the surface or barotropic tide, since it is depth-
independent, is not sensed by the EMVP. Second, work of other in-
vestigators (Hendry, 1974) appears to show that most of the tidal
energy, at least for the semidiurnal tide, is concentrated in the
first two or three baroclinic modes (along with the surface tide).
In terms of the vertical wave number spectra of Chapter II, this
means that any tidal contributions to the spectra are located in
the lowest and, possibly, the second-lowest wave number estimates.
We may then infer that above the first one or two vertical wave
number estimates, the primary contributions to the spectra come
from inertial-internal waves. There is also direct evidence from
iq~
the profiles to indicate that most of the semidiurnal tidal energy
is contributed by waves with small vertical wave numbers. This
evidence will be presented in Chapter V.
We now assume that the observed waves are generated at or
near the surface and propagate their energy downward. Upon reach-
ing the bottom, some fraction of their energy is reflected upward
toward the surface, while the remainder of the energy is lost in a
frictional boundary layer at the bottom. There may also be some
energy loss as the waves propagate through the interior. We also
assume that the smooth bottom does not act as an energy source for
these waves (that is, a boundary layer at the bottom will remove
energy from the incoming waves, but other processes do not act to
put energy into waves that are reflected from the bottom). The
theoretical model presented later in this chapter will hopefully
indicate how reasonable the above assumptions are.
If we accept these assumptions, then the ratio A(m')/C(m')
can be thought of as a reflection coefficient for these waves.
This ratio is shown in Figure 22, using the spectra of Figures 13
and 16. Since the bottom, by assumption, cannot act to put energy
into the waves, a good test of the above model is that A(m')/C(m')
should always be less than one. An examination of Figure 22 shows
that this is substantially true, although at high wave numbers (or
low energy), the ratio A(m')/C(m') behaves erratically. This er-
ratic behavior at low energy levels may be an indication that the
signal-to-noise ratio of the EMVP is becoming small. A low
YI-rr~-wr~ ru~i~ ~--- I-~--~-- ~urr~--c- -- _ n._rri- -~-- --- --~li
I. 2 3 4S / 2m' sdb (x / O )
Figure 22. Reflection coefficient A(m')/C(m').
signal-to-noise ratio will cause the east and north velocity compo-
nents to lose correlation (or coherence). Therefore, the quadra-
ture spectrum of the east and north components will decrease.
Since this quadrature spectrum (in stretched vertical wave number)
is proportional to the difference between A(m') and C(m') (see
Appendix D), the clockwise and counterclockwise spectra will tend
to become equal as the signal-to-noise ratio falls. Also shown
in Figure 22 is the cumulative energy of the stretched vertical
wave number spectrum, expressed as percent of total energy. It can
be seen that about 97% of the total energy is contained in waves
whose ratios of A(m')/C(m') are less than one.
It is important to emphasize that the above interpretation
of the reflection coefficient depends strongly on the assumption
that most of the waves observed in the profiles have frequencies
close to the inertial frequency. The short length of the time
series does not permit good frequency resolution. We have already
pointed out that the lowest wave number estimates appear to be
heavily influenced by the semidiurnal tides.(see Chapter V). At
larger vertical wave numbers it is also likely that waves with
frequencies appreciably greater than f are contributing to the pro-
files. We know that a wave with w greater than f is not circularly,
but elliptically, polarized, with a ratio of semi-major to semi-minor
axes of w/f. The presence of such waves in the profiles would pre-
sumably cause the ratio A(m')/C(m') to increase (the addition of a
circularly polarized wave to one which is strongly elliptically
I~--.
81
polarized results in an elliptically polarized wave). This argument
indicates that A(m')/C(m') is probably an ever-estimate of the re-
flection coefficieft for the near-inertial waves. The reader may
refer to Appendix D for a more complete discussion of the influence
of higher frequency waves on the clockwise and counterclockwise
spectra.
b.) A model for the reflection of internal waves from a smoothbottom
A theoretical model for internal wave reflection can be
developed that supports some of the assumptions made in the pre-
vious section. This model is a variation of one studied by Phil-
lips (1963) that analyzes the reflection of purely inertial waves
from a flat boundary. The present model includes buoyancy effects
but otherwise gives much the same results as Phillips' model.
We start with the usual equations for internal waves, with
buoyancy and rotation:
2u- fv = - P + vV2uat ax
v + fu = - 3R + VIvat ay
aw a- + b + vV2Wat az
l + N2w = 0at
~x +y az . IV-)ax ay + z . IV-l
~II_ I~ j~_jl_______________U__C _I
The reader can refer to Appendix B for definitions of the
'quantities in the above equations. The only additional terms in
the above equations are the frictional terms vV2u, vV2v and vV2w,
where v is the kinematic viscosity coefficient. The Brunt-V'isala
frequency, N, is assumed to be constant. This is not a serious
assumption, since near the bottom where we expect the model to ap-
ply, N is fairly constant with depth.
If all variables are independent of the y coordinate, a
stream function, 9, can be introduced that satisfies continuity:
u = . k- and w = - xaz x. IV-2
If Jp is substituted into equations IV-1, these equations can be
reduced to two equations in p and v:
-V2 a V2) 2+ N f2~ 3 f = O;at at ax ataz
f -_2 + [_-- - V2 ] v = 0 . IV-3az at
The coordinate origin is assumed to be located at the bot-
tom, so that z = 0 there, and the coordinates form a right hand
system. Then equations IV-3 must satisfy the boundary conditions
u(z = 0) = v(z = 0) = w(z = 0) = 0 or:
S= 4~4= v = 0 at z = 0. IV-4az
PII~ I_
We look for solutions of the form:
S= A 6kpz ei(kx-wt)
kiz i(kx-wt)v Ve e . IV-5
If these expressions for i and v are substituted in equations IV-3,
there results a sixth order equation in i:
[R2 ] P6 + [2iw'R - 3R2] P4
+[1 - w"2 - 4iw'R + 3R 2 + i N- R] 02
fw
+[ 2 2 + 2i'R - R2 - i NR] = 0 , IV-6f' f
where w' -and R = -. If we assume that R = 0, the charac-f f
teristic equation for P becomes:
2 N2 - 22 _ f2
or j , = + (N2 - f2) 1/2 IV-71,2 2 - f2
These two roots give the z-dependence that would be expected from
two internal waves propagating without friction.
If we now assume R << 1, we can factor out the above two
roots to get approximate expressions for the remaining four:
~-PI~LY- F-~* SL~,~---_ L X -~II ̂ YI_
2 = - +
R -
(12) - (1N2 - 4' - 2m N2 - 321/2
R2 R wf 2-. _f2 IV-8
If we assume that the - 4 term dominates under the radical, thenR2
the expression for the four remaining roots becomes;
P2 = L (-u' + 1) IV-9
Noting that w is assumed positive, we find that the two boundary
layer roots which give solutions that decay into the interior are
S , i1 -1 (W' r 1)1/2s, 2R IV-10
The four roots of 1 that have been obtained for the two propagating
internal waves and the two boundary layer solutions are then:
1 2 _ f2
- (N2 - 2 1/2.
2 W2 f2
1/2
1/2and ('l)
IV-I1
We note that the thickness of the boundary layer is determined by
the quantity (w' - 1) 1/ 2/iR, for w' positive and close to one. As
e' + 1, or as w + f, the boundary layer penetrates more and more
deeply into the interior.
The expressions for v and i are now:
kkz k2Z k 3z kz i(kx - wt)i= [A e +Ae + Ae + Ae ] e
1 2 3 4
kiz ki 2z ki z ki z i(kx - wt)v =[Ve + Ve +Ve 3 +Ve J e IV-12
where p, through P4 are given above. The A's and V's must be deter-
mined from the bundary conditions at the bottom (equations IV-4).
The terms with V1 and A, in the expressions for 9 and v give the
amplitude of the incoming wave, while the terms involving V2 and A2
give the amplitude of the reflected wave. The average horizontal
kinetic energy for the incoming wave is given by:
Ei = 12 (1 + 2) IV-13
The average horizontal kinetic energy for the reflected wave is:
Eo = IV1 2 (1 + 2) I IV-14
where
A=l R ' I 1 1
(i-l)(m-1)1 / 2 2(i-1) ( -1)/ 2 +1)1/2
IV-15
___UEM~ ____Xn_____h_______~~__l--rx~~_-
and
B = w + i+ (
/2(i-1)
S+ 1
W-1)1/2 (+)1/2
IV-16
and
S= (N2 2
1/2
N2 - m2
Using Ei and E , we can form a
zontal kinetic energy:
E /~ (N2 - W2)/2
Ei (m'2-1)1/2
reflection coefficient for the hori-
+ 1 W' -1+ + O(R).
(-1)1/2 (+l)1/2IV-18 IV-18
This reflection coefficient has several characteristics that
are relevant to the present data. First, although for dimensionless
frequencies w' much greater than one, E /E i ~ 1, as w' - 1 this re-
flection coefficient decreases rapidly (Eo/E i does not limit to zero
as w + f due to the approximations made in the above derivation).
This shows that energy in waves near the inertial frequency may be
greatly reduced as these waves reflect off the bottom. Most of the
energy loss suffered by these waves takes place in the bottom fric-
tional boundary layer (a boundary layer, we recall, that grows in
thickness as w + f). For frequencies close to f, E /Ei is very
IV-17
sensitive to w. Since we cannot resolve w as closely as would be
required to test the above theory, we can only say that the ob-
served "reflection coefficients" (Figure 22) and the possible
boundary layer thickness (described below) can be accounted for if
the frequencies of the observed waves are close enough to f.
Figure 8 appears to indicate that, from 2000 dbar to about
5300 dbar, at least, the average perturbation horizontal kinetic
energy follows closely the average profile of N. This would seem
to imply that the waves are not undergoing strong frictional losses
as they propagate through the water column, and therefore that most
of the energy loss from these waves takes place as they reflect off
the bottom.
There is also a suggestion in Figure 8 that the mean per-
turbation kinetic energy decreases within about 100-150 dbar above
the bottom. This may indicate the presence of a frictional boundary
layer at the bottom. In theory, the horizontal kinetic energy must
go to zero at the bottom. However, the EMVP only measures horizon-
tal velocity to about 10 dbar above the bottom. Thus, it probably
would not see that part of the frictional boundary layer nearest
the bottom, where the greatest energy loss takes place. The error
estimate (Chapter II) calculated for the mean perturbation kinetic
energy profile of Figure 8 indicates that the decrease in energy
near the bottom is significant. We again must emphasize, however,
that this error estimate only takes into account the influence of
instrumental noise.
--lll^--Yn~)l~-DI llP ~L1__~I___~C
Figure 8 also shows an apparent bulge in the mean pertur-
bation horizontal kinetic energy at about 200 dbar above the bot-
tom. This bulge remains even if the kinetic energy profile is nor-
malized by the mean Brunt-V'is l profile (N also increases as the
bottom is approached). The theory presented above predicts only
that the mean perturbation horizontal kinetic energy should increase
smoothly from zero at the bottom to values appropriate to the in-
coming and outgoing waves beyond the frictional boundary layer.
Therefore, this apparent bulge in the kinetic energy near the bot-
tom is not explained by the above theory. The possible existence
of small bottom slopes may be crucial here. The above theory as-
sumes a flat, horizontal bottom. Wunsch (1969) has shown that if
the group velocity vector for the reflected wave has an inclination
angle to the horizontal which is close to the bottom slope, an in-
tensification of the internal wave energy near the bottom can take
place. Since the group velocity vectors for waves near the iner-
tial frequency are only slightly inclined to the horizontal, the
near-bottom behavior of these waves .may depend strongly on small
bottom slopes.
'--- - -I~-~IC PIIPIPlllr~U P WI~-~ir* X -~ I *XI*Ui-I~~.*-III .~LII-_ PrD
Chapter V Comparison of Vertical Wave Number Spectra with Theoret-
ical Models. Computation of Dropped, Time-lagged, Rotary Coherence
(DLRC). The Vertical Energy Flux of Internal-inertial Waves.
a.) Vertical wave number spectra
Recently, several models have been derived which describe
the amount of energy contributed to a region of the ocean by in-
ternal waves of various length and time scales. Perhaps the pri-
mary model of this type is that of Garrett and Munk (1972, 1975).
These authors have constructed an internal wave energy density
spectrum, E(a, w), as a function of horizontal wave number, a, and
frequency, w. The energy spectrum is assumed to be isotropic;
that is, the magnitude of E is independent of the direction of a.
The general form of E(a, m) is (Garrett and Munk, 1972, eq. 6.3):
E(a,w) = C1-A(X)2(w). V-1
This is a dimensionless equation. Time has been made dimensionless
by a reference Brunt-Vdislla frequency,
N0 = 3 cph, V-2
and distance by a reference cyclical wave number,
M = 0.122 cpkm. V-3
C is a constant. p is called the "a-bandwidth of the spectrum" as
___~__Ufl__ X _ _____L____ _~~_li~-~^-
a function of frequency. The parameter X is given by:
A = a/p . V-4
The form chosen for 2(w) was:
= -p+2s 2 -s V-5
where wi is the (non-dimensional) local inertial frequency. The
constants p and s were chosen to be 2 and 1/2, respectively. Then
1/ 2 V-6
In the 1972 version of the model, A(X) was choien to have the form
of a "top hat":
A(X) = 1, for 0 < X < 1
and A(X) = 0, for X > 1. V-7
Finally, p(w) was chosen to be:
() = j.(W- .2)I/2 V-8* 1
where j, is an integer. In the 1972 model, j, was called the
"equivalent number of modes at the inertial frequency" by the
authors. (It should be emphasized that their model does not really
depend on the existence of modes in the vertical. The parameter j,
dan be thought of as characterizing the maximum vertical wave number
found to contain significant energy in the internal wave energy
density spectrum). We would now like to see how this theoretical
model compares to vertical wave number spectra observed by the EMVP.
In order to proceed further we must introduce the dimension-
al forms of the above equations. In what follows, a hat (^) above
a quantity will indicate that it is dimensional. A prime (') will
indicate that vertical wave numbers in the stretched coordinate
are being used.
The dispersion relation for internal waves (under a WKB
approximation) gives:
a = i(z) (2 2 for 2 <)2
A(z) V-9
where ii is the local vertical wave number. Also,
= j*r (W2 i2) .)- = j*r NV-101 N V-O
Then (z) NX = -lJ
= - =
1M (j*) MO (z) V-11
However, since in the stretched vertical coordinate
m' = i(z)(No/N(z)), V-12
we have
(j,) Mo V-13
Then the dimensional form of n(w) becomes
NS(G) = -,
8(A2 _ f2)1/2 V-14
and
N3E(W,m') = C A(X) o
(j.") e(^ - f2) . V-15
The total energy in horizontal velocity is given by
Etotal = E(a,w) WF d~ad V-16
whered2 + W.2
N wo (Garrett and Munk, 1972). V-17
However, since velocity in the profiles has been normalized by
N(z)/N o , we must use
U21
(A(z)/N o ) V-18
instead of U2 in equation V-16. Converting a and w to their
dimensional counterparts, where
A A
a = a/Mo and w = W/N , V-19
we have
E U2' E(a,w) da d' V-20total M0N 0
Instead of integrating over ^, we wish to carry out the integration
over ml. Since
d= 2 - 2)/2 d V-21R(z)
and
d' = (No/i(z)) d, V-22
the integral becomes
Etotal 1 U2' E(i',) (2 1/2 dm ' d
NoMoo M NoA f
Co (,) (-2 +' in' d0 m' 0 W No
S o A( ) dw dm' . V-23(j*7)Mo G 3(-2 - 2)1/2 j Mo
Etotal must be multiplied by PNoMo - 3 to obtain the dimensional
energy spectrum, where P is sea water density (assumed equal to
1 gm/cm3 ). Removing the integral over i' leads to the following
dimensional form of the energy density spectrum in stretched verti-
cal wave number, Mi':
A A CN o ^E(m') = A + f2 dE-,($') = (-M-- ^/)*mIM o j'.M ^ ( _ f2)/2
No2 3Eo _i'
S- A ( )M0 2j*T j. Mo V-24
( C. Garrett, 1974, personal communication).
E is a dimensionless constant which is related to C by
E = C l(w) dw V-25
To obtain the actual energy levels at a given depth, zo Em')
is multiplied by (N(z)/N ). At that level
m' = (zo) (No/N(zo)). V-26
The quantity EA,(m'), as obtained by observations, has been
given already in Figures 12 and 15. Clearly, the form of the
stretched vertical wave number spectrum depends on A(x). In the
1972 model, A(X) was chosen to be a top hat. Comparing equations
V-7 and V-13, we see that A(X) = 1 for m' < j,* THo and A(A) = 0 for
m' > j,Mo. (Again we see that in the top hat model j* controls
the maximum vertical wave number that contributes energy to the
spectrum). This means that the theoretical Em, is independent ofA
m' for X < 1, and is zero for X > 1. It is clear from Figures 12
or 15 that E~, is not independent of ii', but in fact decreases withm
increasing m'.
In a revised form of their model (Garrett and Munk, 1975),
these investigators chose a new form for the function A(X):
A(X) = (t - 1) (1 + X) -t V-27
where t is chosen to fit observation. Garrett and Munk picked
t = 2.5.
1__111___~_______4_1Y_~
V-28
(The choice of t = 2.5 appears to agree reasonably well with verti-
cal displacement spectra obtained by Hayes, et. al.(1975)). For
No = 3 cph, Mo = 0.122 cpkm, Eo = 27 x 10 5 and j, = 6 (in the
1972 model, j. was 20), the vertical stretched wave number spectrum
becomes
2,(i') = 19.1 A(X) (cm/sec)2
m cpkm V-29
and
X = 0.435 i' . V-30
We have repeated the stretched vertical wave number spectrum
of Figure 12 in Figure 23. Figure 23 also shows the theoretical
stretched vertical wave number spectrum, E^.(m'), using the form of
A(X) given by equation V-27, for three choices of j.: j, = 6,
j. = 15 and j, = 20. (The cyclical frequency has been converted
from cycles per kilometer to cycles per sdb). The value of the
constant for the spectral energy level, E(j.), (corresponding to
Eo in the above derivation) has been adjusted to make the three
theoretical curves have the same energy levels at large stretched
vertical wave numbers. If E(j*) is the actual value of the con-
stant used for a given j., and Eo = 21 x 10 5 , we have the follow-
ing values for E(j,): E(j, = 6) = 2 x Eo; E(j, = 15) = 0.6 x E0 ;
E(J. = 20) = 0.4 x Eo. The theoretical curve for a j, of 15
appears to best represent the observed stretched vertical wave num-
ber spectrum, both in the general shape of the spectrum and in the
total energy level.
WAVE LENG TH(SDB)
100 20
0.01 0.05
VER TICA L WA VE NUMBER(C/SDB)
Figure 23. Comparison of theoretical and observed total energy spectrain stretched vertical wave number (see text).\
Kj
(i)
Oz
(3S
(r~
(31
102
10
--lll~li~_~~ I__ ^1 _I*__I.~ I ~ _ I__LIII_1IIYI
A value of 15 for j, is larger than the value of 6 assumed
in the 1975 model. If we return for the moment to a consideration
of Figures 10 and 11, the reason why j, = 6 is apparently too small
becomes clearer. Visually, Figures 10 and 11 show that waves con-
taining relatively high energy have wave lengths as short as
100 sdb. The total depth in the stretched vertical coordinate is
1860 sdb. If we now assume that j, is in some sense representative
of the greatest number of stretched vertical wave lengths that
"fit" into 1860 sdb (and contain significant energy), we find that,
for 100-130 sdb vertical wave lengths, j, should be at least 15.
It is also interesting to note that the value of E required
to make the theoretical (j, = 15) and observed stretched vertical
wave number spectra contain (roughly) the same total energy is
about 60% of the constant, Eo, chosen by Garrett and Munk. The
fact that high-frequency internal waves (those with frequencies
near N) are probably not energetic enough to be sensed by the EMVP
may partially explain this. However, it is also likely that the
total energy of the internal wave field at the MODE central moor-
ing is less than the total energy at other locations (at Site D
(390 20' N, 700 00' W), for example). Since observations at Site D
were used (at least in part) to determine Eo, the fact that
E(j, = 15) is less than E0 may also reflect an actual decrease in
internal wave energy at the central mooring, relative to other
locations in the ocean where measurements have been made.
~I.I_-La~L~XXI~II^WL.1 I~Y~
b.) Dropped, time-lagged, rotary coherence of profiles
The following calculations were carried out in order to
compare profiler horizontal velocity data with some theoretical re-
sults based on the Garrett and Munk model presented in section a.
(C. Garrett, 1974, personal communication). Since we have a time
series of profiles, it is possible to calculate the velocity coher-
ence between.two profiles separated by a time interval, T, for any
stretched vertical wave number. This is called the dropped, lagged,
coherence (or DLC) by Garrett and Munk. However, we know that much
of the variability in the profiles is associated with waves having
periods near the local inertial period. For this reason, instead
of the DLC we have calculated the coherence between two profiles
when the horizontal velocities of the second profile have been
rotated through an angle y:
y = OT,
W = 2rrf, V-31
where f is the inertial frequency and T is the time interval be-
tween the two profiles. This is called the dropped, lagged, rotary
coherence (or DLRC). If the first velocity profile is
W,(i ) = Ul(u ) + iV1(z) V-32
and the second profile is given by
W2 (*) = U2 (*) + iV2( ) V-33
then the coherence is calculated between W1 and
WS = W2e = U3(**) + iV3(i*) V-34
_L---l-~- IIY~ -II-IYII~YX~~IIII IY --̂ Xi-I III-I~L~L~-- -.l11 ~rrrm-~-.
We then obtain the following four quantities:
UU3 PUlU3PUlU3[PUlUl 1 /2[PU3U3]I/ 2 '
PVlV3 = PVIV3
[PVlVI] 1 /2[PV3V3] 1/2 '
QUIU3 = QU1U3
[PUlU] 11/ 2 [PU3U3] 1/2
QV1V3 = QVlV3
[PVIV ] 1/ 2 [PV3V3] 1 / 2 , V-35
where PUIU3 is the normalized cospectrum between velocity compo-
nents U1 and U3, QUlU3 is the normalized quadrature spectrum between
the same components, and PVIV3 and QVlV3 are the same quantities
for the velocity components VI and V3. PU1UI, PU3U3, PVlVl and
PV3V3 are the autospectra for U1, U3, V1 and V3, respectively.
All of the above quantities are functions of stretched vertical
wave number and the time lag between profiles.
We now describe the actual method of computation. Time lags,
T, were calculated for all pairs of drops in the time series. These
pairs were then sorted in such a manner that all pairs with lags
within + 1/2 hour of a given whole hour were assigned a lag equal
to that whole hour. For example, all pairs with time lags of from
2 1/2 to 3 1/2 hours were given a lag equal to 3 hours for the cal-
culation. Calculations were done only for those lags which had
four or more pairs of drops. The results of these calculations are
100
shown in Figures 24, 25, 26 and 27 for lags up to almost two days.
For any lag at which the calculation was made, there are three
points shown for each of the four normalized spectral quantities
given above. Each of these points represents an average over all
pairs available at that time lag. In addition, estimates have
been averaged over several stretched vertical wave number bands.
The number n represents the number of the stretched vertical wave
number estimate; that is, the vertical wave length is given by
1800/n sdb, where 1800 sdb is the total length of the series.
Thus, the three points (at a given lag and for a given quantity)
are averages over stretched vertical wave number estimates n = 1-5,
n = 6-10, and n -, 11-15.
It is somewhat difficult to put error bars on PUlU3, PVIV3,
QUIU3 and QVIV3 since each point may be the result of averaging
over a different number of pairs of drops. In an attemipt to esti-
mate the error we have calculated PUlU3, VlV3, QUlU3 and QVlV3
separately for each pair of profiles in a case when four pairs of
profiles were used to give the estimates shown in Figures 24-27.
(We recall that the smallest number of pairs used was four.) The
resulting values of PUIU3, PVlV3, QUIU3 and QVlV3 have an average
standard deviation about their mean values of about + 0.2. Pre-
sumably, points which are obtained by averaging over more than four
pairs of drops are even more stable; that is, the scatter of points
about their mean values is less than + 0.2 standard deviation.
10
T (HOURS)20 30 40 50
Figure 24. Dropped, lagged, rotary cospectrum (normalized) between velocity componentsU, and U3 (see text for a further description of Figures 24 to 27).
T (HOURS)20 30
Figure 25. Dropped, lagged, rotary cospectrum (normalized) between velocitycomponents V, and V3.
10 40 50
t)
T (HOURS)10
t20 30
* n=I-5o n = 6-IOx n = 11-15
0
xoxg x
Xoo00;6
x x0 o
Figure 26. Dropped, lagged, rotary quadraturevelocity components U, and U3.
spectrum (normalized) between
40 50
<0O
I
L
T (HOURS)20 30
Figure 27. Dropped, lagged, rotary quadrature spectrum (normalized) betweenvelocity components V1 and V .
10 40 50
105
Figures 24 and 25 also show the curve of PUIU3 or PVIV3 ob-
tained theoretically (C. Garrett, 1974, personal communication) from
the Garrett-Munk model under the assumption of an isotropic spec-
trum. The curve for PUIU3 or PVIV3 does not depend on wave propa-
gation in the vertical. In contrast, UlU3 and QVIV3 do depend on
vertical propagation. If only vertical modes exist, QU1U3 and QV1V3
are theoretically always equal to zero. On the other hand, if all
waves are propagating energy downward, the theoretical curve for
QUlU3 or QVIV3 starts out at zero for zero lag and slowly increases
to about + 0.25 for a lag of 50 hours. If we accept the above
"confidence limit" of + 0.2, it is difficult to say whether the ob-
served values of QUlU3 and QVIV3 are significantly different from
zero. Therefore, the following paragraphs will be limited to a dis-
cussion of the observed behavior of PUlU3 and PVIV3. These quan-
tities appear to contain the most interesting information on the
behavior of the DLRC with varying time lag and stretched vertical
wave number.
Using the above error estimate, it is possible to say sev-
eral things about PUlU3 and PVlV3. First, the behavior of PUIU3
or PVIV3 for estimates containing the largest vertical wave
lengths (n = -1-5) is significantly different from the behavior of
these quantities for shorter vertical wave lengths (n = 6-10 and
11-15). The estimates for n = 1-5 decrease much more rapidly with
increasing lag than the estimates for higher n. In fact, the ob-
served PUlU3 and PVIV3 curves for n = 1-5 have an oscillatory
106
character with minima (greatest negative values) for lags of about
12 hours, and again for lags of about 30-40 hours. This is a
strong indication that the longest vertical wave lengths have sig-
nificant energy contributions from waves with periods near 12 hours.
The most obvious candidate for this energy contribution is, of
course, the barotlinic semidiurnal tide. In contrast, the esti-
mates for higher n show little evidence of energy contributions at
approximately 12 hour periods. This implies that most of the
semidiurnal tidal energy is in the longest vertical wave lengths.
Other work (Hendry, 1974) has shown that the baroclinic, semi-
diurnal tide has most of its energy in the lowest three vertical
modes. The above result appears to confirm this.
Second, PU1U3 and PVlV3 for higher n (n = 6-10, 11-15)
follow the theoretical curve more closely. Given the above "error"
estimate, PU1U3 and PVIV3 curves for n = 6-10 and n = 11-15 cannot,
in general, be distinguished. The theoretical carve does not de-
pend on stretched vertical wave number, but the error is probably
large enough so that dependence on vertical wave number cannot be
tested adequately. It is interesting to note that PU1U3 and PVIV3
for n = 6-10, 11-15 are almost always greater than the theoretical
curve. -A possible explanation for this is that while the theoret-
ical curve takes into account all internal wave frequencies (from
inertial to Brunt-Vlisglg), the sensitivity of the profiles is such
that higher frequency internal waves are "filtered out." These
higher frequencies tend to make 1U1U3 or PVIV3 decrease more
107
rapidly with increasing lag than would be true if only near-inertial
waves were being observed. The results of this section may be sum-
marized as follows: 1.) The behavior of PU1U3 and PVIV3 for wave
lengths outside of the region in which semidiurnal tidal energy is
significant does not appear to refute the theoretical calculations,
taking into account the possible sources of error described above.
2.) The result of other calculations, showing that most of the
semidiurnal tidal energy is confined to the largest vertical wave
lengths, is also obtained from the present calculations.
c.) The vertical energy flux of near-inertial waves
We have pointed out that, because of the relatively short
time interval over which the time series was obtained, it is dif-
ficult to obtain good frequency resolution for the observed waves.
However, there are several independent characteristics of the data
which lead us to believe that the waves which we have been calling
"inertial" actually have periods somewhat shorter than the inertial
period.
First, Figures 6 and 7 indicate that the phases of the domi-
nant waves are moving upward in time. If these waves were of ex-
actly inertial frequency, internal wave theory indicates that the
phase should remain at the same depth, independently of time. In
order to propagate upward, these waves must have frequencies greater
than f.
108
Second, consideration of Figure 18 shows that many of the
more energetic velocity features which can be identified between
drops 235U and 236U (these drops being separated by 4.8 kilometers)
do not occur at the same depth in both profiles, but rather are
offset slightly. This depth offset is too large to be accounted
for by errors in the EMVP pressure calibration. The implication is
that the phase planes of these waves are tilted slightly away from
horizontal. Figure 21 then indicates that the wave number vector,
, for these waves must be tilted at a small angle to the vertical,
and equation III-10 shows that the wave frequency, 6, has to be
slightly greater than f.
The third indication that the wave frequencies are somewhat
greater than f comes from attempts to least-squares fit sinusoidal
waves to the data. At any pressure, z, it is possible to fit sinu-
soids of the form:
ufit (Zn,ti) = A(zn) sin((zn) + ti) ,
vfi t ("n,ti) = B(in) sin( (zn) + sti) V-36
over a group of profiles. t i is the time of a given profile. u and
v are east and north velocity components. Then the quantities
U() = . (u (n ,ti) - Uobs (Z ,ti))2 V-37
and
V(n) =i f i t( i) - vob s n'ti))2 V-38
109
give an estimate of the residual error in the fit at depth zn.
These fits can then be calculated over a range of depths, Zn. The
quantity
D = E U() + V(zn) V-39zn
represents a measure of the total residual error of the fit over
the time and depth interval studied. Attempts to do such fits in
the main thermocline, where the strongest signal is found, indi-
cate that the total residual, D, is minimized if the period, T,
used in the above formulas (' = 2r/T) is slightly less than the
inertial period. In fact, the best fit occurs if we choose u so
that T = 22-23 hours. The inertial period is 25.56 hours at the
latitude of the profiler time series.
We have argued in previous chapters that the near-inertial
waves are propagating their energy downward from the surface. Since
waves with frequencies close to f have group velocity vectors that
are only slightly inclined to the horizontal, it is at first glance
surprising that these waves appear to make significant contributions
to the energy even in the deep water. It could be argued that those
waves which have the maximum (downward) vertical energy flux will
be able to penetrate from the surface to the deep ocean most effi-
ciently. This vertical energy flux is given by
F = Cg E V-40
where Cgz is the vertical component of the group velocity vector,
110
and E is the total (kinetic and potential) wave energy. It is of
interest to obtain the local tgz, as a function of z, and use it,
along with the theoretical energy spectrum of the previous section,
to calculate F as a function of frequency and vertical wave number.
Because the derivation of the Garrett-Munk spectrum assumes
that a WKB-type description of the wave behavior is adequate, we
will assume in what follows that this is again true. The equations
to be solved are the same as equations B-6, except that now N is a
function of depth. The equation for the vertical velocity, w, is:
d2dw + 2 2 ,2) w = 0 V-41dz2
with
k;2 +22 =
S - 2 . V-42
To obtain the WKB solution, we let
8 (z)dz 0 A n(z)W= e E n V-43
n=O n V-4
The solutions for n = 0 and n = 1 give, respectively, X and Ao -
2t[a 21/2X = + i [N (z) - w2] V-44
and
= 10 A 2 1/4 V-45
[N -w 2Then
21/2+ i k - 12 2 2 ) dz
1 e- zo2.sE[2(z) - 2]1/4 V-46
111
where the region of integration is over an interval of z for which
N2(z) > ^2. From continuity ,we get u and v:
u = ( ok ifl) 2 1/4 + +B X(z)dzA f(N2(Z) _ 2} 1z/
v = (t iR (2 _ f 2) /2 (k2 + ~2) V-47
The buoyancy is
1 xdz
b= e
i( 2 2) 1/4 V-48
The total energy, E, is given by
* = 1/2 ( < j j2 + I^12 + I -2 )^ ^ ... 1^12 + A12 + 1-1 2 +
2 1/21/4 (N2 - 2) 2 +2 +N2
=21/4 _ ^2 2 2 _ V-49
where the angle brackets denote an average over time. The ratio
of vertical kinetic energy plus potential energy to horizontal
kinetic energy is:
vert. K.E. + P.E. (N2 + W2) (2 f2 )
horiz. K.E. (i - W) (w2 2) V-50
For frequencies close to i, almost all of the energy appears as
horizontal kinetic energy. The vertical component of the group
velocity vector at depth z isgz(8,,z) = = - 2 ^2) (^2 _ 2)
a mu(z) ( - 2)V-51
112
The theoretical spectrum of Garrett and Munk has already been given
(equation V-24). This spectrum contains only horizontal kinetic
energy. However, equation V-50 shows that for frequencies close to
, most of the wave energy is horizontal kinetic energy. From
equation V-24, the energy density spectrum is
N 2
m ( m'',0 ) = BC -- A (--+) +f2Mo3 j,Mo ~ 1M 3(2 _ 2)1/2 V-52
The vertical energy flux for waves in a small interval of frequen-
cy-stretched vertical wave number space Am'AW is
Fz = Cg N (m',) Am' , V-53Fz Cgz NO
where m' and ^ are taken to be in the middle of the element Am'Aw.
The vertical group velocity can be expressed in terms of w and m'
by using equation V-12
^ ^ N ( - f2) (^ - 2)Ceg ( ,m',z) = -
z(z) , (R2 -2) V-54
Then2 1/2NC 2 (W2 + f2)-(N2 w ~2) (W2 2)1/2 #% '
Fz = - C-A M .Mo Mo ^ 2 - 2)
V-55
We note that for w close to f, Fz is almost independent of z.
From the WKB approximation, it can be shown that
U aS= z (Cgz =0. V-56a
113
However, since we are only considering horizontal kinetic energy,
aFz/az is not equal to zero but approaches zero as w approaches f.
We are particularly interested in examining Fz as a function
of w, for fixed m'. The frequency dependence of Fz is given by
g() = (2 + f2) (^2 - ^2)1/ 2(2 - 02)
w (N _ 2
(o2 + f2) ( 2 2)1/2(1 - 2
a' V-57
since N >> f. If W and N are made non-dimensional by
f f V-58
then
(W + 1) (W 2 1)1/2(1 -0)2
2
W V-59
f() is plotted in Figure 28, with N = 5f. It can be seen that
f(') increases rapidly with increasing ' until ' = 1.2, and then
decreases. This implies that, for fixed m', Fz reaches a maximum
for ^ = 1.2f. For an inertial period of 25.56 hours, the vertical
flux of horizontal kinetic energy reaches a maximum (at a given M')
for waves with periods of around 21 hours. This period appears to
be somewhat shorter than that which might have been expected. (We
recall that the least-square fit indicated that the dominant waves
in the profiles seemed to have periods of 22-24 hours in the main
I .0
0.5
0.0-2 3 4 5
Figure 28. Frequency dependence of the vertical flux of horizontal kinetic energy, usinga theoretical form for the energy spectrum of the internal wave field.
115
thermocline.) However, it is significant that the downward energy
flux appears to have its maximum value close to the local inertial
frequency in this model.
A certain amount of inconsistency is inherent in using a
spectrum such as the Garrett-Munk model to calculate energy flux.
The first point of inconsistency, of course, is that the true spec-
trum will not. have an infinity at the inertial frequency. This
problem could be removed by assuming, for example, that there should
really be an energy cut-off near the inertial frequency; that is,
instead of the infinity at the inertial frequency, the spectrum
could be altered locally (at and near f) so that a simple peak oc-
curs at the inertial frequency. This alteration would not change
the flux curve of Figure 28 appreciably. In fact, it would force
the observed flux maximum to be concentrated even more about per-
iods of 21-22 hours. The second problem is more subtle, and in-
volves the question of why one would expect the spectral peak to
be exactly at the inertial frequency. In a case (such as the
Garrett-Munk model) where vertical modes have been assumed to have
been established a priori, it seems that a peak at the inertial
frequency would not be inconsistent, since the waves are not re-
quired to be supported by energy sources located, for example, at
the surface or bottom boundaries. However, once we require that
the waves be supported by a flux of energy into the ocean from the
surface (or bottom) boundary it is not consistent to expect that
the peak in the observed energy spectrum at some depth will be
116
found at that frequency (f) where the vertical energy flux becomes
zero. Both of the above points argue that the "inertial peak"
should actually appear at frequencies somewhat greater than f.
Figure 29 shows a frequency spectrum obtained from averaging
spectra of current meter records at 1500 meters at 12 different
locations in the MODE region (M. Briscoe, 1975, personal communi-
cation). The-average inertial frequency of the 12 locations is
also shown. (The maximum and minimum inertial frequencies of the
12 locations are about + 0.002 cph about the average inertial fre-
quency.) It is clear from Figure 29 that the inertial "peak" is
displaced toward frequencies greater than f. The center of this
wide peak is in the period range 22-23 hours, in agreement with the
above discussion.
It is possible to use the above expression for the vertical
group speed (equation V-54) along with the observed stretched verti-
cal wave number spectrum (Figure 16) to calculate a rough estimate
of the vertical energy flux (downward and net), integrated over all
vertical wave numbers. In the absence of accurate frequency infor-
mation, we have assumed that the clockwise and counterclockwise
spectra are dominated by waves with periods of the calculated max-
imum flux (or of the center of the near-inertial "peak" in Figure
29); that is, periods of 22-23 hours. Using this period range in
the expression for the vertical group speed, and assuming that
roughly half the energy in the clockwise and counterclockwise energy
estimate for the smallest two or three stretched vertical wave
117
.. 00' IO7'cr.
FREQUENCY ICPN)
Figure 29. Frequency spectrum, obtained by averaging 12 spectra computedfrom current meter records at 1500 m in the MODE region. Energydensity units are (cm/sec)2/cph. f is the inertial frequency.
118
number estimates is contributed by the semidiurnal tide (see Chap-
ter V-b), we obtain estimates of the total downward wave energy
flux of 0.6-1.0 erg/cm 2/sec, and of the net downward wave energy
flux of 0.2 - 0.3 erg/cm 2/sec. The estimate of the total down-
ward flux is consistent with other estimates of vertical energy
flux by internal waves (Bell, 1975). We have suggested in pre-
vious chapters that a bottom boundary layer could be a significant
energy sink for near-inertial waves. An order-of-magnitude cal-
culation indicates that a net downward energy flux of 0.2 - 0.3
erg/cm2/sec could be matched by energy loss in a boundary layer.
A measure of this possible energy loss is
L = 4K p((u)2 + (A )2) Az erg/cm2 /sec V-60v Az Az(Yih, 1969)
where Kv is the coefficient of vertical eddy viscosity, p is den-
sity, u and v are horizontal velocity components, and Az is a scale
depth of the boundary layer. Choosing values for Au2 + Av2 and
Az from Figure 8 (Au2 + Av2 - 4 cm2/sec 2, Az ~ 5 x 10 cm), we get
L - (0.3 x 10-2) Kv erg/cm2 /sec. For Ky = 100 cm2/sec, L -~ 0.3
erg/cm 2/sec. Bell (1975) quotes a value for Kv of - 10 cm2 /sec in
the interior but notes that Kv could be at least an order of mag-
nitude larger in a turbulent bottom boundary layer. Thus, the
value of Kv needed to make L comparable to the estimated net down-
ward energy flux is not inconsistent with other estimates of Kv.
119
Chapter VI Some Observations over Rough Topography
In previous chapters we have presented an analysis of data
obtained over smooth topography (at the MODE central mooring).
Some of the calculations presented, such as reflection coefficients,
depend on the assumption that the bottom is not acting as a wave
source. Over the smooth topography this does not appear to be an
unreasonable assumption. However, it is well known from theoretical
studies that a rough bottom may interact with either a mean flow,
producing lee waves, or with an oscillatory flow such as a tide.
Bell (1975) has discussed the possible generation of internal waves
in the open ocean by rough topography. In an earlier work,
Hendershott (1964) analyzed a case where inertial waves could be
generated by the oscillatory flow of a diurnal-tide component over
a rough bottom. His study indicated that strong inertial wave
currents might be generated at latitudes slightly to the south of
the critical latitude (the latitude at which the tidal and inertial
periods are equal). The critical latitude for the 24-hour diurnal
tide would be 300 N, for example. His theory actually predicts
that there should be a somewhat complicated band structure of the
inertial energy south of the critical latitude, with a primary peak
in the inertial energy just south of the critical latitude, and
secondary maxima farther south.
120
It is important to realize that there are numerous tidal
lines that are called diurnal.' Although the critical latitude for
the 24-hour tide is 30* N, the critical latitude for the 24-hour
50-minute tide moves down to about 28.90 N. This means that the
actual energy levels of the inertial waves (generated by topography)
might not depend as strongly on latitude as is predicted by the
theory. In any case, the fact that the major diurnal tidal compo-
nents at 280 N have frequencies within the internal wave frequency
range, but near the inertial frequency, indicates that it would be
worthwhile to look for near-inertial energy propagating away from
the bottom in the rough topography region.
During the MODE experiment, a set of prcfiles was obtained
around a ridge in the rough topography region east of the central
mooring. These drops are indicated by a dagger (t) in Appendix A.
The center of the ridge is located at approximately 280 04' N,
68* 31' W. The main part of the ridge is oriented along the axis
NNE-SSW and rises about 700-800 meters above the adjacent, smoother
part of the ocean floor. Because this set was obtained in order to
make a survey of the velocity structure around the ridge it is not
a time series at one location. In fact, the set is made up of a
series of pairs of profiles, except in a few cases where only single
profiles were made. Each pair was obtained at one location around
the ridge, and the second profile was usually taken at about one
half of an inertial period after the first profile. Because a time
series of profiles is not available over the rough topography, it
121
is difficult to accurately remove the low-frequency shear profile.
For this reason, the results of calculating clockwise and counter-
clockwise vertical wave number spectra are more difficult to inter-
pret than are the corresponding results over the smooth topography.
However, some significant features in these rough topography pro-
files can be seen visually; that is, without recourse to spectrum
calculations.
Figures 30 and 31, for example, show hodograph plots (plots
of the horizontal velocity vector with depth) for profiles 190D and
192D between 3000 dbar and 4500 dbar. These two profiles were made
at a time lag of one half of an inertial period. The original pro-
files have been smoothed by applying a running 150 dbar triangular
filter to each velocity component. This makes it easier to vis-
ually follow the behavior of the longer wave length (higher energy)
waves in the profiles. Smoothed points are plotted every 10 dbar,
and depth is marked every 100 dbar. The most striking feature of
these hodographs is that they show wave-like structures for which
the horizontal velocity vector is clearly rotating counterclockwise
with increasing depth. The structure is clearly time-dependent, as
can be seen by comparing the hodographs for profiles 190D and 192D.
Comparison of the hodographs appears to indicate that these waves
have periods near the inertial period. We have already pointed out
in Chapter III that such counterclockwise polarization indicates
that these waves are propagating energy upward, away from the ridge.
NORTH (CAM/SEC)0ro)
CO'(V
0NO
|V
t)|
0.
-
L)6 I
1 L.
38
37
34
40
32
3630
4543
3.0-
2.5-
2.0-
1.5-
1.0-
0.5-
0.0-
-0.5
Figure 30. Smoothed hodograph of profile 190D between 3000 dbarand 4500 dbar. Depths are in hundreds of decibars.
4.5
122
4.0 -
3.5-
lLu
K)K;(I)
p 1-
O
d.
EAS T (CM/SEC)I
0U' 06 001 I0
-t r IrI. _
34
33
5.0
4.5-
4.0-
3.5-
3.0-
2.5-
2.0.
30
31
1 I I I I I I I I
Figure 31. Smoothed hodograph of profile 192D between 3000 dbarand 4500 dbar. Depths are in hundreds of decibars.
5.5
123
32
K)
Q::(I)
1.,5-
i.U
124
Although spectral estimates obtained from only two profiles
are probably not very stable, it should be mentioned that vertical
wave number spectra (without stretching the vertical coordinate)
obtained in the deep water (2000 dbar and deeper) for drops 190D,
1910, 192D and 193D.show a consistent dominance of counterclockwise
energy over clockwise energy. These four drops were made at the
southwest end- of the ridge.
In contrast, deeper water (greater than 2000 dbar) spectra
and hodographs for a drop pair, 195D and 197D, that was made north-
east of the ridge, do not show any strong tendency for counter-
clockwise rotation with depth. If anything, these drops show a
tendency for the velocity vector to rotate clockwise with depth in
the same manner as was observed over the smooth topography. Swallow
floats located near the ridge when these profiles were made indi-
cated that the mean flow (at 4000 dbar) was to the southwest.
Since the pair 195D and 197D was located to the northeast of the
ridge while 190D, 191D, 192D and 193D were to the southwest, we
might speculate that if the ridgewas generating near inertial waves,
these waves would be advected to the southwest and therefore would
be seen primarily by profiles to the southwest of the ridge. Al-
though it is difficult to draw general conclusions from a small set
of profiles, the above observations in the deep water around the
ridge can be summarized as follows: profiles made to the southwest
of the ridge show clear evidence of counterclockwise polarization
with depth; profiles made near the ridge crest (such as 189D, for
125
example) do not appear to show a dominance of either polarization;
two profiles (195D and 197D) made over a smoother region to the
northeast of the ridge appear to have reverted to the clockwise
polarization with depth characteristic of the smooth topography
profiles, at least for the longer vertical wave lengths, which are
most energetic.
Bell (1975) has calculated a value for the possible upward
energy flux of internal waves generated by rough topography of
-~ erg/cm 2/sec. This value is comparable to the total downward
flux in the smooth topography region, as estimated in Chapter V.
If the average vertical energy flux values for bottom topography
and sea surface generation are comparable, it might be expected
that energy input at the bottom could sometimes exceed energy input
at the surface (especially since both input processes are probably
intermittent in time). This would result in waves polarized
counterclockwise with increasing depth, at least in the deep water
over the rough topography. (We have also examined hodographs of
the rough topography profiles in the upper half of the water
column. In contrast to the plots in the lower half, these hodo-
graphs do not show a clear dominance of one polarization over the
other, indicating that in the upper half of the water column a
complicated addition of the top and bottom energy sources may be
taking place.)
126
Chapter VII Conclusions, a Further Discussion of Some of the Data
and Recommendations for Future Experiments
The previous chapters have been devoted to an analysis of
internal wave data obtained by vertical profiling. It is now
worthwhile to summarize the significant results of this analysis
and place these results in the general framework of internal wave
theory.
One of the more fruitful results of this analysis has been
the ability to relate the observed characteristics of wave polar-
ization and vertical phase propagation to the locations of energy
sources for these waves; that is, to the possible existence of
internal wave energy sources at the top and bottom boundaries of
the ocean. As Muller and Olbers (1975) have pointed out, a two-
sided, symmetric, vertical wave number spectrum (equal energy in
positive and negative vertical wave numbers) is not consistent with
the existence of net boundary sources at the surface and bottom.
A major cause of difficulty with internal wave observations until
now has been the inability to observe asymmetry of internal wave
vertical wave number spectra (or, in fact, vertical wave number
spectra) of horizontal velocity with the techniques that have been
employed. Time series of horizontal velocity obtained from current
meters at fixed locations in space cannot distinguish between an
internal wave having an upward or a downward phase propagation,
127
since the horizontal velocity vector rotates in the same sense for
either wave (clockwise in the northern hemisphere). The tech-
nique, employed in Chapter II, of decomposing vertical profiles
into clockwise and counterclockwise energy contributions is a
useful method for observing asymmetry in the internal wave vertical
wave number spectrum and for inferring the direction of net verti-
cal energy flux from this asymmetry. A similar instance where
vertical wave polarization has been used to infer the direction of
vertical energy propagation in the atmosphere has already been
pointed out (Hines, 1966). A case where vertical polarization of
tidal currents was apparently observed in an analysis of current
records obtained from current meters on moorings has been reported
by Titov and Fomin (1971). We have mentioned in Chapter III, how-
ever, that the ability to identify clockwise and counterclockwise
energy with waves having upward and downward wave number vector
components (that is, with the two-sided vertical wave number spec-
trum) requires that the observed energy be primarily contributed
by waves with frequencies near the local inertial frequency. This
is true because individual waves are nearly circularly polarized if
their frequencies are near the local inertial frequency. If a
single wave with a frequency much greater than inertial were pres-
ent, for example, this wave would provide energy contributions to
both the clockwise and counterclockwise vertical wave number spec-
tra, since the wave would be elliptically polarized in the vertical.
The clockwise energy would still be greater than the counterclockwise
128
energy if the wave were propagating energy downward. But the
true, two-sided vertical wave number spectrum of this wave would
contain energy only on one side of the spectrum (at the positive
or negative vertical wave number of the wave, depending on which
way the phase of the wave is propagating in the vertical).
Fortunately, the data presented and analyzed in previous
chapters indicates that the two dominant sources of energy in the
profiles are near-inertial waves (at all vertical wave numbers),
and the semidiurnal tide (at the smallest vertical wave numbers).
Inertial and diurnal tidal contributions to the profiles cannot,
however, be resolved because of the short length of time over which
the profile time series was obtained. The ave-age frequency spec-
trum at 1500 meters (Figure 29) shows that a large part of the energy
observed at this depth in the internal wave band comes from fre-
quencies near the inertial-diurnal tidal frequency, and from the
semidiurnal tide. The observed "mirror-imaging" of profiles (Fig-
ure 3) and the calculations of dropped, lagged, rotary coherence
(Figures 24 to 27), support this.
The observation that much of the variability in the profiles
was.caused by the near-inertial waves (and the semidiurnal tide at
the largest vertical wave lengths) led to efforts to evaluate quan-
titatively the total and net downward energy flux over the smooth
topography. Making assumptions about the.amount of energy contri-
buted by the semidiurnal tide at the smallest vertical wave numbers,
we obtained values for the total vertical energy flux downward of
129
~ .6 - 1.0 erg/cm2/sec, and for the net vertical energy flux down-
ward of ~.2 - .3 erg/cm 2/sec. Although both estimates are subject
to uncertainty because of the presence of the semidiurnal tide,
neither estimate seems unreasonable (see Chapter V-c). In partic-
ular, it appears that the net downward energy flux can be balanced
by a rough estimate of the rate of energy loss per cm2 in the bot-
tom boundary layer (if we assume, of course, that the observed de-
crease of mean perturbation horizontal kinetic energy near the
bottom is actually caused by the presence of a boundary layer).
In relation to the contour plots (Figures 6 and 7), a fur-
ther statement can be made which, although somewhat-obvious, is no
less important. The observed relations among wave period, direc-
tion of vertical phase propagation, and vertical wave polarization
in Figures 6 and 7 constitute at least a partial verification of
the dispersion relation for internal waves (those near 'the inertial
frequency). Other observations and analyses of data (from current
meter records, for example) have assumed that the observed oscil-
lations between the inertial and Brunt-Viisdl' frequencies obey
the dispersion relation for internal waves. In the absence of clear
evidence of the propagation of oscillations in this frequency
range, other investigators (Webster, 1969, for example) have ar-
gued that some other process (such as turbulence) could explain
the observed oscillations. Although turbulence is difficult to
define, one of its characteristics is that it does not follow a
well-defined dispersion relation. In this sense, then, the results
130
of Figures 6 and 7 clearly argue for internal waves as being the
cause of the observed variability in the profiles, at least for
the near-inertial waves.
Another significant result brought out by this analysis is
the fact that both profiler data and other, independent observa-
tions (Figure 29) have shown that the energy concentration in what
one might call the "inertial peak" is, first, not concentrated in
a well-defined peak (as is the semidiurnal tide, for example) and
second, is not concentrated at exactly the inertial peak but is
rather centered at a frequency about 20% greater than the local
inertial frequency. Some previous observations have also shown that
the inertial "peak" is quite often displaced toward frequencies
somewhat greater than the local inertial frequency. Webster (1968),
for example, shows a frequency spectrum of current at 38* 28.8' N,
70* 00.5' W, in 3300 meter-water, with the observations made about
30 meters above the bottom. This spectrum has a clear concentra-
tion of energy near the local inertial frequency, but the center
of this peak occurs at a frequency about 5% higher than the local
inertial frequency. White (1972) considered temperature data ob-
tained in the upper several hundred meters at a site in the Pacific.
His temperature spectra always show an inertial "peak" at frequen-
cies up to 30% higher than the local inertial frequency.
Various attempts have been made to explain this shift theor-
etically. Munk and Phillips (1968), for instance, argued that
internal waves generated to the south of a given latitude of
131
observation (in the norther hemisphere) would propagate northward,
and that as they propagated northward their frequencies would ap-
proach the local inertial frequency. A test of this idea by the
use of profile: data would require better information on the direc-
tion of horizontal phase propagation for the observed waves. We
will return to this question later, when the possible influence of
the mean vertical shear on the propagation of the waves seen in
the profiles is discussed.
White (1972) pointed out that the observed frequency shift
of the inertial waves could be explained as the result of a
Doppler shift due to the mean flow. Because he always observed a
shift toward higher frequencies, he was forced to conclude that
the near-inertial waves he observed were always propagating in the
direction of the mean flow. The data presented in previous chap-
ters, however, show that it is unlikely that Doppler shifting could
be the cause of the observed frequency shift. The average fre-
quency spectrum at 1500 meters (Figure 29) shows that there is an
appreciable (20%) shift of the "inertial peak" toward higher fre-
quencies. But it is known from a calculation of the average flow
observed by a current meter at 1500 meters near the location of the
five-day time series (Chapter II) that the mean flow at that depth,
when the time series was made, was small (< 3 cm/sec). It does not
seem that such a small mean flow could account for the observed
frequency shift. This does not necessarily negate White's conclu-
sion, since he was observing at a point where the mean flow was
132
about five times that observed at 1500 meters in the present exper-
iment. It would seem fortuitous, however, that the near-inertial
waves would always propagate in the direction of the mean flow,
especially if, as the present data indicate, they are geing gen-
erated by atmospheric forcing at the ocean surface. In fact, there
is a suggestion in the data (discussed below) which indicates that
at least the more energetic features in the profiles have a compo-
nent of propagation to the north. Since the mean flow, at least to
about 1500 meters, was to the south during the present experiment,
this would imply that, if anything, the inertial peak would be
shifted to lower frequencies.
The results of the profiler observations now allow us to
propose another mechanism to explain why the observed "inertial
peak" frequency shift almost always appears to be toward higher
frequencies. This explanation is that the observed waves are
clearly propagating in the vertical. The contour plots (Figures
6 and 7) show that the dominant structures associated with the
near-inertial waves are moving upward in time, corresponding to
downward energy propagation. (Of course, these contour plots are
influenced by the mean flow. However, we have already pointed out
that around 1500 meters the mean flow is small, and yet the great-
est speed of upward phase propagation in these contour plots is
found in the 1500 dbar to 2500 dbar range.) In order to propagate
vertically, these waves must have frequencies greater than the
local inertial frequency. The calculations of Chapter V-c showed
133
that, using an assumed form of the internal wave energy spectrum,
the maximum vertical flux of horizontal kinetic energy occurs at
frequencies about 20% higher than the inertial frequency. We
pointed out that, once the energy sources are placed at the surface
(or bottom) boundary, it is inconsistent to have the peak in the
energy spectrum appear exactly at the inertial frequency when waves
of this frequency cannot propagate into the interior. It is con-
sistent, however, to have an "inertial peak" that is shifted to
higher frequencies. It seems likely that the waves which, when
generated at the surface, have the greatest downward energy flux
would be most able to penetrate through the total water column.
(They would presumably be able to travel a greater distance in the
vertical before losing an appreciable fraction of their energy due
to friction, non-linear interactions, shear interactions, or some
other cause.)
We have argued that a significant part of the energy put in
by the atmosphere is lost in a bottom boundary layer, at least for
the near inertial waves. Calculations of the net downward energy
flux and the rate of energy loss in what we are interpreting in
Figure 8 to be a bottom boundary layer show that (in terms of a
rough calculation) the energy flux into the boundary layer could be
accounted for by the energy loss in the boundary layer. It is cer-
tainly true that other factors can act to remove energy from the
wave field. For example, it is clear that the observed waves tend
to increase in amplitude and decrease in vertical wave length in
134
the thermocline, where the Brunt-Viisili frequency is large. For
waves shorter than some critical value, the shear caused by these
waves will be large enough so that the Richardson number would be
less than 1/4, and the flow caused by the wave might become unstable.
This would extract energy from the waves (possibly putting that
energy into turbulence). Such a process would be most active at
the high-vertical-wave number end of the vertical wave number spec-
trum. It is also reasonable to expect that frictional losses will
be important throughout the water column (not just in a bottom
boundary layer) for waves with short vertical wave lengths. As the
vertical wave length decreases, the frictional term; v 32 /dz2, in
the horizontal mrmentum equation (Chapter IV) for near-inertial
waves grows. In terms of the boundary layer discussed in Chapter
IV, we can say that (assuming the frequency is fixed and close to
inertial) as the vertical wave length decreases, the horizontal
wave length will also decrease. This will cause the parameter
R = vk2/f of Chapter IV to increase. When this occurs, frictional
losses are no longer confined to a thin bottom layer, but rather
become important in the interior of the water column as well. This
indicates that the argument that near-inertial waves are losing most
of their energy in a bottom boundary layer is probably only appli-
cable for near-inertial waves with small vertical wave numbers.
This process of energy loss due to friction in the interior of the
water column would also be expected to be most active at the high-
vertical-wave number end of the vertical wave number spectrum.
___L__XIL~___I I;__C- .-̂I-I~1IYII~-LII~-I~ -~i~lL
135
Another cause of energy loss by these waves would be by
interaction of the waves with the mean flow. In processing the
profiler data we chose to normalize the profiles by the mean Brunt-
Vais5li profile. This type of normalization was used because it
made the calculations easier to compare with existing theories.
However, it is well known (Phillips, 1966) that a vertical shear
in the mean horizontal velocity profile can also cause changes in
the amplitude and wave length of internal waves as these waves
propagate through the shear profile. Perhaps the greatest diffi-
culty in evaluating the influence of the shear on the profiles is
lack of data on the direction of horizontal phase propagation of
the waves relative to the mean shear flow. We have made several
attempts to determine the direction of horizontal phase propaga-
tion. One attempt involved calculating the coherence between temp-
erature and horizontal velocity components as obtained by the pro-
filer in individual drops. In theory, observed phase differences
between temperature and east velocity component and temperature and
north velocity component, in the case of a single dominant wave,
would allow one to determine the direction of horizontal phase
propagation. In practice, the observed levels of coherence between
temperature and the two velocity components were too low to infer
anything about horizontal phase propagation direction. Using the
observed energy levels of the near-inertial waves in the main ther-
mocline and the mean vertical temperature gradient observed there,
a rough calculation indicates that a near-inertial wave with a
136
frequency 20% greater than the local inertial frequency would pro-
duce a temperature signal that would be measurable in the profiles,
given the resolution level attainable by the EMVP temperature meas-
urements. The fact that the coherence between temperature and
velocity measurements is so low may be due to the presence of
temperature variations caused by waves with frequencies much greater
than the inertial frequency (the semidiurnal tide, for example).
In order to use the temperature profiles to determine horizontal
propagation direction, better frequency resolution of the profile
data must be obtained. Comparison of drops 235U and 236U (Figure
18), both visually and by doing lagged correlations between the
east components and north components of the two profiles, suggests
that strong features which can be identified between the two pro-
files occur at a slightly greater depth in profile 236U (4.8 kilo-
meters to the north of 235U). If this is true it implies that
these waves have a component of propagation toward the north.
Taking into account the fact that the mean shear flow is primarily
in the north-south direction (Figure 7), it would be expected
(Frankignoul ,1972) that a wave group with a component of propaga-
tion to the north and propagating energy downward would be acted
upon by the shear in such a way that the vertical wave length would
decrease with increasing depth. In fact, if the wave is thought to
be propagating directly north (against the major part of the mean
shear), an analysis of the WKB expressions for the change in verti-
cal wave length brought about by changing N and the presence of the
137
shear suggests that the two influences could be of comparable mag-
nitude. However, Figure 3 shows that the most energetic waves have
wave lengths in the deep water that are longer relative to observed
wave lengths in the thermocline. We have also tried to use the
stretched, normalized profiles to see if there is a shift of ener-
getic spectral peaks in the upper versus lower parts of the
(stretched) water column. We chose a set of stretched profiles
(230D through 233D) which contained a distinct peak at about
130 sdb. Vertical wave number spectra for the upper and lower
halves of these profiles were then calculated separately. On the
assumption that the stretching of these profiles has removed
changes in vertical wave number and wave amplitude due to variations
of N, any change in the position of the peak in the upper half rela-
tive to the lower half might reflect the influence of shear. The
result showed that there is a slight shift toward higher stretched
vertical wave number in the lower half relative to the upper half,
but it is very small ( < 5 sdb). One might expect a shift in this
direction if the waves have a component of horizontal propagation
to the north. The fact that the shift is so small might indicate
that the waves are propagating primarily east-west, with a small
propagation component to the north.
The major conclusion of the above paragraphs is that a good
evaluation of the influence of the mean shear on wave propagation
awaits better estimates of horizontal phase propagation directions.
138
A final important result of the present work is that wave
structures have been observed over the rough topography which have
polarization appropriate to upward energy propagation. As pointed
out in Chapter VI, there is a theoretical basis for believing that
rough bottom topography at the latitude of this experiment ( ~ 28°N)
could generate near-inertial waves that must propagate their energy
upward, away from the bottom. Bell (1975) has calculated that
rough topography could support an upward energy flux of - 1 erg/cm 2/
sec. If we accept our estimate of .6 - 1.0 erg/cm 2/sec downward
for the energy flux to internal waves input by some (as yet only
poorly understood) atmospheric forcing, then we see'that rough
topography generation of upward energy flux can be comparable to
atmospheric generation of downward energy flux. Some data from
current meters at 4000 meters in the MODE region (D. Porter, 1975,
personal communication) indicates that, in the band of periods from
37 hours to 14 hours (encompassing the inertial period), the total
energy in the frequency spectra obtained at 4000 meters depth over
the smooth topography is significantly less (at least by a factor
of 2) than the total energy in the above frequency band in spectra
obtained at the same depth over the rough topography. The above
observations suggest that, at least in regions of rough topography,
one might sometimes expect to see a convergence of energy toward
the thermocline, due to the combined effects of atmospheric and bot-
tom forcing. According to Frankignoul and Strait (1972), such a
convergence in the near-inertial frequency band can be seen in
139
records obtained at Site D (390 20' N, 700 00' W). We again empha-
size that we have not seen any evidence of dominant upward energy
propagation (i.e. counterclockwise polarization with depth) in the
smooth topography time series.
The above discussion makes clear some of the problems asso-
ciated with interpreting the present analysis. These problems, in
turn, suggest some future experiments which could be performed in
order to clarify the influence of various processes on internal
wave propagation. We would now like to propose several relevant
experiments. First, a time series, similar to the one obtained
over smooth topography, should be performed over a rough topography
region. One of the primary difficulties with interpreting vertical
wave number spectra of deep water profiles over the rough topography
is that the low-frequency shear profile cannot be accurately removed
if only two profiles have been obtained at a given location (at a
time interval of one half of an inertial period). Any remaining
low-frequency components to the profiles will cause errors in the
calculated magnitudes of the clockwise and counterclockwise spectra.
Although the smoothed hodographs of Figures 30 and 31 are clearly
rotating counterclockwise with depth, and the main features in the
hodographs appear to reverse direction over the one-half inertial
period time interval between the two profiles, it is difficult to
put any confidence in the interpretation of spectra calculated from
single or paired profiles. A time series over rough topography
could result in other interesting observations as well. For
140
example, one might speculate that the observed semidiurnal tidal
contribution to the profiles, which resides in roughly the lowest
five stretched vertical wave number estimates at the smooth topog-
raphy observation side, might show a change of character over the
rough topography. Interaction between the rough topography and
"low-mode" semidiurnal tidal waves might lead to the scattering of
the "low-mode" energy into higher modes. It would be interesting
in this regard to compare the DLRC calculations (Figures 24 to 27)
done for the smooth topography time series with the same calcula-
tion done for a time series over the rough topography. If, as
seems likely, the bottom-induced generation of near-inertial waves
is intermittent in time or space, then the "quick-look" capability
associated with the EMVP data processing system would be an in-
valuable aid in determining when and where to perform the rough
topography time series.
A second experiment involves using a pair of EMVP's to per-
form repeated paired profiles, such as those shown in Figure 18.
The single pair in Figure 18, as already pointed out, has shown
that there is an apparent tilt to energetic features that seem to
correspond to each other in the two profiles. Repeated paired
profiles would allow one to make a much more certain estimate of
this tilt and would also (if series of paired profiles were made
along two orthogonal directions) allow one to make an estimate of
the horizontal wave propagation direction.
_1~1_~ __j~_l~_~__~_~_
141
A final possible experiment would involve the combined data
of an EMVP time series performed at some location (at the central
mooring, in smooth topography, for example) along with a current
meter mooring at the same location which is densely instrumented
for the first several hundred meters off the bottom. This exper-
iment would both help to verify the indication in the present EMVP
data that there is a layer near the bottom in which the average,
high-frequency, horizontal kinetic energy appears to decrease
toward the bottom and would also provide frequency information
which, as indicated in Chapter IV, is crucial in determining the
rate of energy loss undergone by near-inertial waves as they
reflect off the (smooth) bottom.
Perhaps the usefulness of a set of experimental observations
can best be measured by whether analysis of that set of data clearly
illuminates the way along which more comprehensive theoretical and
experimental work may be carried out. The present analysis has
generated both significant results and significant questions which
must be answered by further work, and, therefore, under the above
criterion this set of observational data has been extremely useful.
a 142
APPENDIX A
Profile Locations and Times
Drop
177
178
179
180
181
182
183
184 t
185 t
186 t
187 t
188 t
189 t
190 t
191 t
192 t
193 t
194 t
195 t
196 t
197 t
Date 1973
April 10
" 10
" 10
" 12
" 12
" 12
May 14
" 15
" 15
" 15
" 16
" 16
" 16
" 17
" 17
" 17
" 17
" 18
" 18
" 18
" 18
Drop Time (GMT)
1627
1738
1858
1539
1658
1833
1458
1423
2045
2238
0430
1115
1750
0029
0547
1305
1842
0132
0801
1418
2044
Position
32021'.0N 64034'.5W
32021'.ON 64034'.5W
32021'.0N 64034'.5W
32021'.0N 64034'.5W
32021'.0N 64034'.5W
32021'.0N 64034'.5W
29024'.ON 69039'.8W
28001'.5N 68030'.OW
28001'.9N 68032'.8W
28002'.7N 68033'.8W
28001'.4N 68030'.3W
28003'.2N 68034'.0W
28002'.9N 68030'.9W
27056'.1N 68044'.5W
27057'.9N 68039'.8W
27056'.1N 68044'.3W
27058'.ON 68039'.OW
28008'.2N 68028'.3W
28"08'.0N 68018'.OW
28008'.5N 68028'.5W
28008'.6N 68018'.3W
143
Drop
198 t
199
200
201
202
203
204
205
206
207
208 t
209 t
210
211
212
213
214
215
216
217
218
219 *
220 *
221 *
Date 1973
May 19
" 22
" 22
" 23
" 24
" 24
" 24
" 25
" 25
" 25
June 1
" 1
(" 8
1" 8
" 9
S" 9
5" 9
" 9
5" 9
" 10
" 10
" 11
" 11
" 11
Drop Time (Gt4T)
0231
0045
1612
1512
0737
1454
2034
0310
0915
2126
0204
0559
2257
2257
0530
1217
1217
1829
1829
1220
1220
0523
1152
1815
Position
28*02'.5N 68033'.6W
27006'.6N 69040'.12W
28000'.ON 69019'.OW
27043 ' . 2N 70001'.8W
27059'.3N 69039'.OW
27057'.ON 69036'.2W
27059'.2N 69038'.OW
27059'.1N 69038'.4W
27058'.ON 69038'.0W
29021'.7N 70042'.2W
23002'.ON 680 30'.8W
28002'.ON 68030'.9W
27023'.5N 70001'.2W
27023'.5N 70001'.2W
27029'.8N 7000'.4W
27023'.5N 70001'.2W
27023'.5N 70001'.2W
27029'.8N 70000'.5W
27029 '.8N 70000'.5W
27039 '.5N 69031'.0W
27039'.5N 69031'.0W
27059'.8N 69039'.1W
27059'.7N 69038'.9W
27059'.8N 69039'.2W
Drop
222
223 *
224 *
225
226 *
227 *
228 *
229
230 *
231 *
232 *
233 *
234 *
235
236 *
237 *
238
239 *
240 *
241
242 *
243 *
244
245 *
Date 1973
June 11
" 12
" 12
" 12
" 12
" 13
" 13
" 13
" 13
" 13
" 14
" 14
" 14
" 14
" 14
" 14
" 14
" 14
" 15
" 15
" 15
" 15
" 15
" 15
Drop Time (GMT)
1815
0043
0714
1301
1945
0219
0759
1220
1800
2117
0102
0451
0828
1210
1206
1630
2019
2016
0034
0438
0438
0800
1219
1219
144
Position
27*59'.9N 69039'.0W
27059'.7N 69039'.3W
27059'.8N 69*39'.2W
27052'.0N 69039'.2W
27059'.8N 69039'.0W
27059'.8N 69039'.1W
27059'.8N 69039'.2W
27051'.6N 69042'.9W
27059'.7N 69039'.0W
27059'.7N 69038'.6W
27059'.8N 69038'.8W
27059'.8N 69039'.1W
27059'.8N 69038'.9W
27057'.6N 69038'.7W
27 059'.8N 69039'.IW
27059'.8N 69039'.1W
28003'.6N 69042'.0W
27059'.9N 69038'.8W
27059'.4N 69039'.1W
28005'.7N 69044'.7W
27059'.8N 69039'.1W
27059'.7N 69039'.1W
28000'.6N 69039'.8W
27059'.9N 69039'.0W
_____I*I~__/UYI*C___L~I^II~_.~~~II1~^- PL~I^-ZXI-
Date 1973
June 15
" 15
Drop Time (GMT)
1935
2057
Position
27052'.1N 69039'.OW
27052'.2N 69038'.8W
(t) rough topography drops
(*) time series drops
246
247
145
_L________1__111_____Y___X ~_a~____ _YX~ I ~I___I___ ILI3*I__LU__.I
146
APPENDIX B
Use of Electro-magnetic Velocity Profilers
to Measure Internal Waves
As was pointed out in Chapter I, a conversion equation must
be used to obtain horizontal water velocity from values of horizon-
tal electrical current density recorded by the EMVP. This conver-
sion equation involves not only the above variables, but also the
vertical water velocity at the point of measurement, as well as the
horizontal gradient of any electrical potential that exists in the
water. Vertical water velocities and horizontal potential gra-
dients, therefore, introduce errors into the calculation of hori-
zontal water velocity. We will now solve a rather simple problem
that allows us to obtain an estimate of this error.
The model chosen is that of a rotating ocean with constant
Brunt-Vaisalg frequency and constant depth, H, and with a rigid
surface and bottom. We assume that internal wave vertical modes
exist in the water column. This assumption is not crucial in esti-
mating the error, as will be seen later.
The equation for the electrical potential, 9, and the elec-
trical current density, S = (Jx, Jy, Jz), generated by an ocean
moving with velocity 9(x,y,z,t) = (G,vw) is:
A A
V$ x F -J/a , B-l
147
where a is the conductivity of the sea water, which is assumed to
be constant, and t = (0,F y,Fz) is a vector representing the terres-
trial magnetic field. The coordinates x, y, and z are, respectively,
magnetic east, magnetic north, and vertical. The coordinate origin
is located at the sea surface, and z is positive upward.
Equation B-1 will be called the full conversion equation,
since it contains all terms that relate $, v and J. The horizontal
components of equation B-1 are:
Jx-=F Fz F -
a Z y ax
3-Y- -0iF - _a z ay B-2
The underlined terms make up the "applied" conversion equation, or
the equation that is actually used to convert EMVP electrical cur-
rent readings to horizontal water velocity. The remaining terms
cause the error. (The reader should refer to Sanford, et. al.
(1974) for a more comprehensive discussion of EMVP measurements.)
Since V * S = 0 and V x F = 0, the equation to be solved
becomes
V20 = V (v x F) = F (V x ) B-3
with the boundary conditions:
148A
w = Jz = 0 at z = 0;
S= Jz = 0 at z = -H B-4
Evaluating x F, these boundary conditions may be rewritten:
W(z = 0) = w (z = -H) = 0 ;
~ (z = 0) = Fy * u(z = 0) ;az y
B-5AA (z = -H) = F ^u(z = -H) .
az y
Given (x,y,z,t) we can, at least in principle, solve for P and
then 3.
The internal wave equations are:
AU A
at _ f a2kat ax
axax
=- P+az
+ N2w = 0
v +aw+ -- +B-6
where u, v and w are, respectively, the east magnetic, north mag-
netic and vertical velocity components, f is the Coriolis parameter,
149
and p is pressure divided by a reference density, o. We have used
the Boussinesq approximation, in which the density, ^ , is assumed
to be given by three terms:
P = Po + W(z) + p(x,y,z,t), B-7
and the water is assumed to be incompressible. The water pressure
is made up of two parts, pop and PoP. The hydrostatic part of the
pressure is given by:
P d(z) - g(P + (z)), B-8dz
where g is the acceleration due to gravity. Then poP is the devia-
tion of the pressure from hydrostatic, the bucyancy, b, is -gp/po
and the Brunt-Vais*l* frequency, N, is:
N= ( -B1 -9Sdz
By Fourier transforming equations B-6, where
i(kx + ly -wt)u(x,y,z,t) = u(z;k,l,w) e , B-10
and similarly for the other variables, we obtain the following
equation for w:
d2w + (k2 + 12) 2 2 = 0. B-idz2 W2 - f2
The form of w that satisfies the boundary conditions on vertical
velocity at the surface and bottom is:
niTz
A sin Hn7T
H
With n an integer, and
n2r2 2 2 N2 )22 I (k2 + 12) N2 _ W
H2 w2 - f 2
Equations for u and v are:
N2 _ 2
iw(W2 _ f 2 )
N2 - 2
iW(W2 _ f 2 )
nrrzcos H
n7tH
cos H
nr -
H
(ilf + kw);B-14
(Iw - ikf)
B-15
- (k 2 + 12) F * (V x )
nirz
F sin HY nit
H
+F cos Hz nit
HH
(AN2 2 (ilf + kw) - ikA
iW(w2 _ f2)
Bf(k 2 + 12)
B = - -AL -2 2
nL iWm( 2 - f2 )Ti
150
B-12
B-13
Au -nit
H
AnTV
Then
dz2dz-
where
B-16
B-17
151
The boundary conditions on are then:
S F ,B(ilf + kw)dz = n at z = 0 8-18dz nw
Hand
F yB(-l)n(ilf + kw)dz n at z = -H . B-19
H
The system of equations B-16, E-18 and B-19 can now be
solved to give:
Y=A N -~( ( i f - i k ) - i k+ n 2 + k2 (2 - f
SFy A(N2 -w 2 ) (if - ikw)
/f +72 W(W2 f2)( )t
x coth 42 + 12 H - (- 1 )n csch + 12 H x cosh 2 z
(A2 2+ 2 2( 2 f2 ) (If - ikw)-ikn k2 2 W(W2 f2)
FyA(N 2 _ 2) (If- ik) sinh X 2 +2 z
W(W2 _ f2)()2 V + 12
H
n7ir
FyA 2 ,2) (If - ikw) - ik
n2 2 +k 2 2 nH
- -H
152nwz
F Af(N2 - 2) (k2 + 12) n'rzz Cos H+ i
- W(2 _ f 2 )HB-20
The full conversion equations have been given above (equa-
tions B-2). The applied conversion equations (those actually used
to convert from ) to F) are:
X (v -V) Fz , B-21
Jy -u )Fz
aB-22
where If and v are depth-independent terms (Sanford, 1971).
Since we are only interested in the part of J that varies with
depth, we set
= v =0.
If we now let utrue and vtrue be the velocity components astrue true
given by equations B-14 and B-15 and u and v beobs obs the compo-
nents calculated from the applied conversion equations B-21 and
B-22, then:
Utrue - Uobs = z 1 ay-Fz a y B-23
true obstrue obs 1F( + wF y)
We see that the expressions for 4 and w can be used to estimate the
error in the conversion equations. At a given depth, z, the energy
B-24
153
in the u error signal is given by:
Eu (z) 1/2 < (U u- u u )* >error true obs true obs
2Fz ay B-25
where the brackets < > indicate an average over time. Similarly,
the v error signal is given by:
Ev (z) = 1/2 < ( v ) ( -v ) >
error true- obs true obs
2 < + w F 122Fz ax y B-26
We note that:
Eu (z) = 1/2 < u * u >true true true B-27
and
Ev (z) = 1/2 < * * v >true true true . B-28
Returning momentarily to the expression for 5 (equation
B-20) we see that the first two terms are proportional to
cosh /k2 + 12 z and (sinh (42 + 1 z)/ k2 + 12). We now make the
assumption that the horizontal component of the wave number vector,
S+ 12, is small enough so that.the cosh term is independent of z
and the sinh term becomes a linear function of z. Since we are only
154
interested in that part of 4 that is depth-dependent, we ignore
the hyperbolic cosine term. The term (sini , T + 12 z)/( 2 + 12)
z as 2 ' 2i - O0. If we examine the coefficient multiplying
(sinh 2 + 12 z)/( 2 + 12) we see that for k2 + 12 << (n2 T2)/(H2)
the last term in the coefficient cancels the first term, and we are
left with (-i kF A)/( n ) as the coefficient of the term propor-y H2
tional to z. For k = 0 this term disappears. Even if k 0 this
term decreases rapidly as n increases; that is, for higher modes or
for frequencies close to f. Since we are only interested in those
terms that are sinusoidal in z, we will ignore the terms in ¢ that
are either independent of z or are only weakly dependent on z.
Then nrz
2:- 2 2 + k2 A 2FA (If - ikw) - ikj sin Hn + k 2 + 12 2) nH 2 +
-Y-- ~H
F Af(N 2 - W2) (k2 + 12) z---
n7 W(W2 _ f2) n_
H H B-29
(The reason for this assumption has been explained in Chapter II.
Any depth-independent contributions to the profiles examined in
Chapter II have been removed prior to analysis. Thus, the (approx-
imately) depth-independent contributions of 4 should not appear in
the profiles.)
We are particularly interested in the relative error in
uobs and v obs' given by Eu rror/Etrue and Ev /Everror true error true
155
respectively. For simplicity, we have averaged the quantities
Euerror , Ev error, Eu , and Everror error true tru
Then
true H ) true0
over depth, z, from 0 to -H.
true > dz ,true
B-30
and similarly for the other three quantities. The final result is
that Eu /Eu and Ev /Everror true error true
Eue= rror
Tutrue
( 22 k
2
X k2 2 + 12f 2
( )1 HIf
F2n2 r2+z H2Hk2 + 12)
W2K2(W2 _ f 2 )
(N2 - 2) (k2 W2 + 12f2) +
are given by:
a2 - f2
F N2 - 2
F 2f2(k2 + 12)z
[t'' *lfZ(~ 2 I:(k2 + 2f2)( 27)2
H
Everror
Tv -Evtrue
x (12f 2 + k2W2
(H2n) 2
H
(n 2 +k2 + 12)2(12 + k2f2)
H
+ 2wk'( 2 - f2 ) + k2(W 2 - f)
(N2 _ 2) (N2 - W2)2
2W2 (W2 _ f 2 )2( + k2 + 12) F 2
(N2 -W2)2 F 2
( N 2 . 2 ) 2 F 2
(nH2 + k2 + 12)2w2(w2 - f 2 )H2
(N2 - W2)2
f2k2(k2 + 12)2+nT 2
It
B-31
F 2
B-32
h
156
Considering first the u-component error, it can be shown
that Tu is a maximum, for a given w and n, if the wave is propa-
gating in the north-south direction (k = 0). Further, for waves
traveling in the east-west direction 1 = 0, and Tu = 0. For
1 n <<(n), we find that max 12
12 <(<()2, we find that max {Tu} 1 , if the frequency, w,nir 2
is close enough to f so that (w2 - f 2 )/(N 2 _ ,2) is small. Since
12 -2 _ f2
H 2 B-33
from equation B-13,
W2 _ f2max {Tu} a
N - 2 B-34
Equation B-33 or B-34 shows that the relative error in the u-compo-
nent energy depends only on frequency, w, or (what amounts to the
same thing) on the north-south "aspect ratio" of the wave, 1/(n/H).
Similar statements may be made about relative error in the
v-component energy. In this case, Tv is a maximum, for a given w
and n, if the wave is propagating in the east-west direction (1 = 0).
However, Tv does not go to zero as k goes to zero, because of the
presence of the term involving vertical velocity in the full conver-
sion equation for v. If (W2 _ f 2 )/(N2 _ 2) is small we find that
max (Tv} o((nw/H) B-35
157
As was the case for Tu, max {Tv} for small (w2 - f2)/(N2 _ W2)
depends only on the frequency, or alternatively, on the east-west
aspect ratio k/(nr/H).
Using values of the Brunt-VMis&lN and inertial frequencies
appropriate to the MFODE center at a depth of 4500 dbar, we find
that the relative error terms max {Tu} and max {Tv} are both less
than 5% for periods down to about one-half day. This range of
periods from inertial down to one-half day contains roughly three
fourths of the energy observed in internal wave spectra from the
MODE area.
We have ignored vertical variations of N and the fact that
the observed waves are propagating in the vertical. Since the
error terms are proportional to (w2 - f2)/(N2 - w2 ), and this
would not change if waves were propagating vertically, the presence
of vertically propagating waves would not change the error estimate
significantly.
The influence of varying Brunt-Vaisal frequency is somewhat
more difficult to determine. If N varies weakly with z, this means
the forcing terms on the right hand side of equation B-17 have
coefficients that are weak functions of z. If we assume that N
varies on a scale that is appreciably longer than the scale of the
waves and try to solve the forced problem using two depth scales,
we would find that the lowest order estimate of the relative error
terms would again be proportional to (W2 - f 2 )/(N 2 _ W 2 ) but that
now N is a weak function of z. The value of N used above to get
158
relative error estimates was taken to be that at 4500 dbar because
this value is close to the minimum average value in the N profile
and thus presumably maximizes the relative error.
159
APPENDIX C
Accuracy and Precision of Observations
It is important to distinguish between accuracy and preci-
sion when interpreting velocity measurements obtained by the EMVP.
This is true because of certain characteristics of the EM1VP it-
self. A detailed description of the arrangement of sensors on the
EMVP can be found elsewhere (Sanford, et. al., 1974). It has been
pointed out (Sanford, 1971) that the electrodes that sense hori-
zontal electric current density only measure the depth-varying part
of the current. That is, there is a depth-independent part of the
current that is not sensed by the EMVP. Therefore, the instrument
measures horizontal current in much the same way that geostrophic
shear calculations do, when a "level of no motion" must be assumed.
This is clearly a loss of accuracy in the instrument.
A second cause of loss of accuracy in the EMVP involves the
actual location of the electrodes on the instrument. As the pro-
filer falls, it rotates. This rotation causes an oscillatory cur-
rent to flow in a coil within the profiler, since the coil is
rotating in the terrestrial magnetic field. This oscillating cur-
rent is then used to determine the point in a cycle of rotation
when the coil is pointing toward magnetic north. The orientations
of the two independent sets of electrodes on the profiler are de-
fined relative to the direction of this "compass coil." In the
present configuration of the instrument, one set of electrodes is
160
oriented parallel to the compass coil, and one set is oriented
perpendicular to it. However, these "perpendicular" and "parallel"
orientations relative to the compass coil can only be measured to
about a degree. The result is that errors in defining the orien-
tation of the electrodes on the instrument cause errors in the
determination of the magnetic bearing of the horizontal electric
current density vector, and thus errors in the measured water ve-
locity. This is a systematic error, which is another cause of loss
of accuracy in the measurements. If, in a small section of a pro-
file, velocity readings from the two independent electrodes are
compared, it is usually found that although the two velocity read-
ings track each other closely with changing depth, there can be a
constant offset between the two of up to 1 cm/sec. If this offset
is removed, the actual RMS difference between the two readings is
usually less than 0.5 cm/sec. This RMS error can be taken to be
an estimate of the precision of the velocity measurements.
A more detailed discussion of EMVP velocity measurement
errors can be found in Sanford et. al. (1974). The above sources
of error should be kept in mind when interpreting velocity profiles.
As mentioned in the introduction, the EMVP also carries
pressure, temperature and conductivity sensors. We will consider
the pressure sensor error first. Because data in the EMVP is re-
corded in digital format, there is a quantizing error in any re-
corded variable. In the case of pressure, this quantizing error
amounts to about 1.5 dbar. Bottom pressure values as obtained by
161
the EMVP for each drop in the time series have been compared in
order to estimate the precision of pressure measurements. If we
assume that the bottom at the time series site is flat, so that the
EMVP hit bottom at the same depth in each drop, then a comparison
of bottom pressures as recorded by the EMVP indicates a root-mean-
square error of about + 1 dbar at about 5500 dbar. This is a meas-
ure of the precision, or repeatability, of the pressure measurements.
On the other hand, accuracy is limited by the fact that values of
pressure computed from PGR (Precision Graphic Recorder) readings
could be determined to only about 5 dbar.
Temperature and conductivity measurements were notably less
successful than pressure measurements. The quantizing errors for
temperature and conductivity are 0.0170 C and 0.02 mmho/cm. The
quantizing error in temperature is twice what it should have been,
since the least significant bit in the digitizer was usually stuck
in the "1" mode. In addition, due to a failure of the conductivity
head on the profiler (the quartz liner in the head cracked under
high pressure and fell out), the conductivity measurements of the
head were seriously influenced by temperature and pressure. This
made the computation of salinity difficult. Although a curve was
derived which corrects conductivity for the above influences of
temperature and pressure on the head, the resulting computed salin-
ity was only accurate to about 0.035 o/oo. This was not judged to
be sufficient for density computations, and so these measurements
have not been used in the present work.
L~YYI___~___III__CIP___CII_____IY- .
162
Finally, a comment should be made on navigational accuracy
during the time series. Ship positions were determined by Loran C.
Although nominally performed at one location, a comparison of re-
corded ship positions for all drops during the time series indicates
that the ship positions have a standard deviation of about + 230
meters (east-west) and t 270 meters (north-south) around a mean
position of 27* 59.75' N, 690 39.04' W. It should be noted that
the above numbers are based on ship position at the beginning of
each drop. The EMVP usually spent 5 to 10 minutes on the surface
before it began its descent. -During this time the instrument was
drifting free of the ship and was presumably carried to the south
by the mean flow (see Figure 5). This additional drift of the
instrument to the south would result in a shift of the mean posi-
tion of the time series to the south (probably by about 100 to 200
meters) but presumably would not increase the standard deviation of
the drop positions appreciably.
163
APPENDIX D
A More General Treatment of the Partition of Energy
Between Clockwise and Counterclockwise Spectra
We let m be the vertical wave number in the stretched verti-
cal coordinate, f be the inertial frequency, and w be the wave fre-
quency. In what follows, we will use the vertical coordinate s,
where s = -z . Thus s is a stretched vertical coordinate increas-
ing upward. The horizontal velocity components of a single internal
wave may be represented by:
2ri(ms - t) y 2ni(ms - t)U (s,t;w,m) = e ; U (s,t;w,m) = -e , D-1L T W
where UL and UT are the horizontal velocity components parallel
and perpendicular to the direction of horizontal phase propagation,
respectively. Then (Garrett and Munk, 1972), the east and north
(u and v) velocity components are given by:
u = i UL cos - U sinO
and
v = i U sin + UT cos , D-2
where 4 is the horizontal phase propagation direction. We adopt
the convention that m is always positive. A wave with upward phase
propagation then has a positive frequency, w, while a wave with
downward phase propagation has a negative frequency. All
164
contributions to the profiles from upward propagating waves may
then be represented by:N
SC 2ni(ms - at)UV (s,t;,) = ) A(w,m;) e dm dw
f 0
andN C
f (2 i(ms - wt)U (st,;) = A(w,m;4) e dm d D-3
7 D-3f 0
The amplitude function, A, is considered to be non-zero only for
f < w < N and m > 0. Therefore, the double integrals can be con-
sidered to be integrals from -o to +co. We similarly represent the
contributions to the profiles from downward propagating waves as:
0O 00
U0 (s,t;) = f (,,m;) e 2ni(msdm dw
and00 O
U0 (st;) f= B(w,m;o) e dm dwT ' D-4
In this case, B is non-zero only for -f < w < -N and m > 0.
The total horizontal velocity components are then given by:
uoU= U V+ U and U= U+ U . D-5
The cross correlation between u and v and the autocorrela-
tion for the u and v components are:
165
p (.,t;)6l(T) -<< u(s,t)v*(s + ,t + T)
+ u (s,t)v(s + ?,t + T) 6(t)6(t - T) > D-6
and
[p (S,T;4) + p (-,T; )] 6(T)
= << u(s,t)u*(s + T,t + T) + u*(s,t)u(s + T,t + T)
+ v(s,t)v*(s + -,t + T) + v*(s,t)v(s + T,t + T) 6(t)6(t - T) > t>
D-7
where the angle brackets denote averages over s and t. The 6-func-
tions arise because we are making a linear section of the random
wave field along the s axis (Konyaev, 1973). (The contributions
to u and v at a given frequency and vertical wave number are
assumed to be a sum of waves with random phases.)
We assume that the above integral limits may be taken as - c
to + c. This implies an infinitely deep water column. Strictly
speaking, the limitation to a finite depth will cause the vertical
wave number spectra calculated below to be smoothed by a window
which is a function of (stretched) vertical wave number. The finite
depth limitation has been ignored in what follows. Using the above
expressions for the correlation functions, we can calculate the
cospectrum (P), quadrature spectrum (Q), and the total energy
spectrum (TE) for zero time lag. These quantities are given by:
166
P(m;4) + iQ(m;4) = P (s,T; 4 )6 (T)e 2i(m ds dT
D-8and
TE(m;¢) =
-0 -00
(P, + P-27i(ms - wT) d
)(S-,T;4)6(T)e ds dT
D-9
Carrying out the required calculations, we find:
P(m;) + iQ(m;4) =
-N
+ f - f -
-f
f
1/4 IcossinOf
N
(1 - -)IA(w,m;c)j2dwW
jB(w,m; )12 dw
f IA(w,m;) 12 dw +
D-10
and
TE(m;) = 1/4 IA(m,w; )12 (1 +
f2(1 +(I + -- ) dw
D-1 1
We can simplify these expressions if we assume:
f2 ) dw
X~_ ~_~ __I(Il*_S11I____Y__Xitll^l ll~ - _1
f IB(o,m;)j'12 dW
JB(W,m;4) 12
167
IA(m,w;)l 2 (1 - a(m,w'))E(w')A(m)R(4) ;
jB(m,w;4)j2 = a(m,w')E(w')A(m)R( ), D-12
where w' > 0. The value of a(m,w') is between zero and one. Use
of this coefficient allows us to indicate how much energy is con-
tributed to a one-sided frequency spectrum by waves with upward
and downward phase propagation. R(4) is assumed to be normalizedA
so that its integral around 2i is one. E(w') and A(m) could be
identified with the frequency and vertical wave number dependence,
respectively, of the Garrett-Munk spectrum given in Chapter V.
Integrating around 4, we get
N
TE(m) 1/4 E(')A(m) ( ,12 ) dw' 0-13
f
andN
Q(m) = 1/4 E(w')A(m) ( (1 - 2a) dw' D-14
f
The clockwise and counterclockwise spectra are given (to
within a constant) by:
C(m) = TE(m) + 2Q(m) ;
A(m) = TE(m) - 2Q(m) . (Gondlla, 1972). D-15
We point out that neither TE nor Q contains multiplicative factors
involving cos or sin4 in equations D-10 and D-11. These quantities
(and therefore C(m) and A(m) do not depend on the distribution of
~YY II-_ll-sX~~-l~lll__lI_ _ i-.-~I l^-LIIIIIC-~L~
168
R(O) around a circle. This is .another way of saying that C(m)
and A(m) are invariant with respect to a rotation of the hori-
zontal coordinates. If we now assume that E(W') has a sharp peak
at a frequency w' near the inertial frequency, so that w'/f ~ 1,O 0
then
A(m) 1 - a(m, wo)
C(m) a(m, "4) D-16
which is the ratio of downward to upward energy propagation. For
a small a (which means that most of the energy is propagating down-
ward), A(m) is much larger than C(m). It is important to remember
that s is positive up. If we were to use z , the stretched verti-
cal pressure coordinate which increases downward (as for the spec-
trum in Figure 13, for example), then C(m), the clockwise spectrum,
would dominate. (Changing the vertical coordinate from s to z is
equivalent to inter-changing A and C.) Several other points can
be made. First, if a = 1/2 everywhere, the quadrature spectrum will
be zero, and clockwise and counterclockwise energy will be equal.
The choice of a = 1/2 corresponds to the case when, on the average,
equal energy is contained by waves with upward and downward energy
propagation. Second, if an energetic wave is present with a fre-
quency much greater than inertial (such as the semidiurnal tide),
we might expect that the addition of this wave will cause the ratio
A(m)/C(m) to approach one. It is interesting to note in this con-
text that the estimates of A(m)/C(m) (using z as the vertical
--̂ -L~plCUrYI11.~31~(P--LI I~-~--~Pq LI-~ -~- .I_.__I~LI~--~l
169
coordinate) at the smallest vertical wave numbers in Figure 22
(where we know that the semidiurnal tide is important) do seem to
increase, relative to estimates at neighboring, higher stretched
vertical wave numbers.
----"i .~~~cll~ssr^l-------~~-rr~-u~-l *ayyu -
170
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_LL--I-I 11.I-I^^^Il X~E~IY
174
BIOGRAPHICAL NOTE
NAME: Kevin Douglas Leaman
BORN: November 23, 1948, Lakewood, Ohio
MARITAL STATUS: Married to the former Diane Saul
EDUCATION:
Institution
Univ. ofWashington
Univ. ofMichigan
EMPLOYMENT:
Institution
Univ. ofMichigan
Woods HoleOcean. Inst.
Dates ofAttendance
9/66-6/67
9/67-6/71
Dates ofEmployment
9/69-6/70
1/74-6/75
DegreeEarned
B.S.M.S.
Date ofDegree
12/7012/72
Major Field
Oceanography
OceanographyOceanography
Duties
Teaching Assistant in Departmentof Meteorology and Oceanography
Research Assistant inPhysical Oceanography
PUBLICATIONS:
Translation from Russian to English of Konyaev, K. V. SpectralAnalysis of Random Processes and Fields. Moscow: "Nayka" Pub-lishing House. Microform Publication MP0001 from AmericanGeophysical Union, Washington, D. C., 1973.
Leaman, K. D. and T. B. Sanford, Vertical energy propagation ofinertial waves: a vector spectral analysis of velocity profiles,Jour. Geophys. Res., Vol. 80 (in press), 1975.
